|2016 electoral college map (source: Wikipedia)|
The electoral college right now is
First Alternative: Direct Vote
The most obvious solution to the above problem(s) is to take out the middleman: down with the electoral college! This approach would require virtually no changes on the ground (just a pesky constitutional amendment). It would be pretty simple to just keep everything as it is (that is, individual citizens vote for the name of a candidate for President) and just count who gets the most votes overall. Elections can still be run by the states as they are now (which I think is very good because it makes it a lot harder to tamper with elections at scale), and it would also match what regular people expect: I vote for Clinton, she gets one vote; my neighbor votes for Trump, he gets one vote. No more of this sleight-of-hand we have now where even though I may punch a ballot next to the name "Hillary Clinton," I'm technically voting for some faceless Democratic Party hack who in turn will really vote for her--describing this process is itself almost enough to condemn it based on weirdness alone.
So elections would look the exact same to voters, there just wouldn't be any Electors who are invisibly elected and who then go vote mindlessly for the candidate they represent at the sham electoral college. Easy-peasy. Plus, a direct vote would also have the benefit of being more democratic, and who doesn't want that?
Second Alternative: A Reinvigorated Electoral College
Well, I'm not sure I want more democratic. I might actually lean towards my second proposal, which would be to make a radical change to how things are done today (but which, ironically, would be close to a return to how things were originally done). In any event, it's more fun to think and blog about this alternative because it's much more complicated :)
Basically, if we don't want to move to a direct vote, let's resurrect the electoral college into the deliberative body it was meant to be!
Original Intent and How it Died
The original idea of the electoral college, as explained by Hamilton in Federalist No. 68, was to have us local citizens elect a trusted smart person to go to a state gathering of elected trusted smart people and have a debate about who to vote for for President, and then all vote for who they think would be best--opinions of the ignorant populace be damned! As Hamilton put it, the choice of who should be President ought to be made by people "most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations." In other words, the electoral college was designed specifically to keep out demagogues like Donald Trump who every thinking person realizes is entirely unfit for the job.
Unfortunately, political parties quickly figured out they could have their partisan hacks run for the electoral college and then all vote for the same dude. And states wanted to maximize their power by having all their votes go to the same person, which meant having statewide slates of Electors all of the same party that voters could choose. So the system soon evolved into pretty much the one we have now. 
A Halfhearted Solution
Alexander Hamilton did not like these developments, so he drafted and proposed a constitutional amendment to both keep the voting for Electors at the district level (not winner-take-all for the whole state) and also to fix that other weird fact that the Vice President is the person who gets the second-most votes for President, potentially creating a party split in the executive (this latter part was adopted as the Twelfth Amendment). But Hamilton's amendment still wouldn't stop parties from running their own guy in each district, pledged to the party's preferred candidate. Such a system, then, would probably result in a better (as in more democratic) version of our current system where each congressional district, more or less , gets its own electoral vote. It would be the same as if we kept the same system we have now except with 535 states that each had their own single vote for President instead of 50 states (plus DC!) that each had different numbers of votes for President.
This would no doubt be an improvement since it would be a few steps towards taking out the problematic electoral college middleman, but it wouldn't be likely to make the Electors exercise any more independent judgment. So if this approach appeals to you, why not just go all the way and make it a true direct vote instead?
A More Radical Approach
No, to get back to a truly independent body not made up of Electors mindlessly voting for their party's pre-selected candidate, we have to be even more radical, in the literal sense of that word: getting back to the root, the original. The Framers meant for Electors to be a deliberative body of thoughtful people who considered who would be best to lead the nation. These stewards were to be elected by the people, yes, but also to exercise their own wise judgment. To get to that ideal, it would obviously require a constitutional amendment, and I think it would need at least three main sections: one restricting presidential campaigning, one ensuring that Electors be independent, and one requiring that Electors be chosen locally. 
One of the Framers' electoral assumptions that broke down fastest (damn you, Burr!) was that campaigning for President would (continue to) be considered unseemly and so no one would do it. Thus, I think the first and biggest thing this hypothetical amendment would need to do is prohibit people from running for President! Yes, that's a big change, but hear me out.
Here's something like the text I have in mind: "No person who has publicly expressed the desire to be President, or who has been nominated by their political party to be a candidate for President, in the last five years shall be eligible to the Office of President." 
But why would we get rid of presidential elections? For the same reason people often say that anyone who aspires to be President should automatically be disqualified. Or, as Douglas Adams put it:
The major problem—one of the major problems, for there are several—one of the many major problems with governing people is that of whom you get to do it; or rather of who manages to get people to let them do it to them.
To summarize: it is a well-known fact that those people who most want to rule people are, ipso facto, those least suited to do it.
To summarize the summary: anyone who is capable of getting themselves made President should on no account be allowed to do the job.
To summarize the summary of the summary: people are a problem. We could (at least begin to) get around that by ending open campaigning for the presidency. No more would there be (due to our inevitable two-party system) a binary choice between the two richest/charismatic people who yearn for power: the Electors could consider anyone they wanted in an environment (relatively) free from politicking!
Of course, people who really wanted to become President could privately tell their influential friends to go out there and talk them up as potential candidates in the media and such; at some point there's not much you can really do to take ambition out of the equation. But at least those subordinates couldn't openly state that they were speaking on behalf of the candidate, because that would just be having an agent publicly express the candidate's desire to be President, which would disqualify them. And it seems to me that you would end up with a decent number of "favorites" in the mix for each party, like how for 2020, Joe Biden, Cory Booker, Kirsten Gillibrand, Kamala Harris, Bernie Sanders, and Elizabeth Warren (and that's just off the top of my head) are all presumed to be interested in the Democratic nomination even though none have announced their candidacy. And of course no matter how many high-profile names get themselves talked about in the media, the Electors could always pick somebody else entirely.
Electors Can't Pre-Pledge
Relatedly, I would want a section to make sure that Electors haven't pre-pledged to anyone. In other words, I don't want an Elector-candidate to publicly pledge that they'll vote for some popular political figure in order to gain votes. Taken to the extreme, a political party could just get its Elector-candidates to all publicly pledge to support Joe DiMaggio and then basically campaign on his behalf in their district. I would also want to guard against political parties deciding in secret who they want their nominee to be and only running their Elector-candidates who would support them, so the pledging need not even be public to be disqualifying.
This section might read something like: "No person shall be eligible to be chosen as an Elector if they have declared for whom they plan to vote for the Office of President, and every Elector shall be free to vote according to their conscience."
Local Elections of Electors
And then we could basically add the first section of Hamilton's proposed amendment (mentioned above), which required each Elector to be selected by a vote of the local district's citizens:
Congress  shall from time to time divide each State into Districts equal to the whole number of Senators and Representatives from such state in the Congress of the United States, and shall direct the mode of choosing an Elector of PresidentThis was/is necessary (1) because the current Constitution technically allows for each State to choose how their Electors will be appointed, and for the first few national elections some States had their legislatures pick them; and (2) in order to counteract the prisoner's dilemma situation that encouraged States to choose their Electors via a statewide winner-take-all election so as to maximize the State's power.  With these changes in place, we would be assured that each district would choose their own representative, not just cast a vote towards a full slate of statewide Electors who would all be of one party and/or pledged to one party's candidate.
and Vice Presidentin each of the said Districts, who shall be chosen by Citizens who have the qualifications requisite for Electors of the most numerous branch of the State Legislature, and that the districts shall be formed, as nearly as may be, with an equal proportion of population in each, and of Counties and, if necessary, parts of Counties contiguous to each other, except when there may be any detached portion of territory not sufficient of itself to form a District which then shall be annexed to some other part nearest thereto. 
All Together Now
Putting it all together, the overall effect of the no-running-for-President section, the no-pledging section, and Hamilton's local-election-of-Electors section would (hopefully) be that we could choose smart people who would go into the electoral college with a relatively open mind about who they would vote for.
To be clear, I don't think it's feasible (or wise) to try to keep partisan politics out of the electoral college altogether--that would take a huge restructuring of our entire government and First Amendment rights. Instead, I imagine that Elector-candidates will run in their district-wide elections  with their party affiliation listed on the ballot, presumably after a primary, so that it would be similar to electing your House representative now: you can vote in the primary if you're really motivated, otherwise you can just show up in the general election and vote for your party's Elector-candidate. I also imagine local debates between the Elector-candidates from different parties that individual citizens can follow and participate in. Candidates for Elector wouldn't be allowed to name a specific person they'd vote for, of course (no pre-pledging!), but they could explain the characteristics that they think are most important in a President and/or the policy positions that they think it is most important for the President to pursue. And if there's a sitting President of an Elector-candidate's party who is eligible for reelection, I could imagine a norm arising where if the Elector-candidate doesn't explicitly say "I won't/would prefer not to vote for the current President to be reelected" then it can be assumed that they'll vote to reelect them. (Note that this would be permissible because it's not publicly declaring who the Elector-candidate will be voting for, but rather who they wouldn't vote for; or perhaps an incumbent President could be an exception to the prohibition on pledging--again, details to be worked out.)
As for how the actual electoral college would go down, I would assume that the Electors of the party with the most Electors (assuming they constitute a majority of all Electors) would want to coordinate among themselves who to choose so that they don't end up splitting their votes and letting the other side end up winning (or at least getting enough votes to throw it to the House). Thus, in the period between Election Day and the time the Electors cast their ballots, the Electors would be free to meet, talk, and/or communicate amongst themselves, and they would undoubtedly do so, probably separated by party. I would also allow the Electors to reach out to the people they're considering voting for at this point to ensure they'd be willing to accept the nomination. 
Because they'd be coordinating nationally anyway, it might also make sense for all of the Electors to cast their votes in one place rather than in their own state. I would love to see that televised, like the old-school party nominating conventions where there was honest-to-goodness intrigue rather than a pre-determined outcome! The voting should also be open ballot so that you can decide if the Elector you chose did a good job in case they run again in four years.
And if you really want to spice things up, you could even make it so that Electors from the losing party could offer to form a coalition with the moderates from the opposing party to form a majority to elect a more centrist member of the winning party. I don't think I'm a fan of this idea since it might lead to only radically centrists getting elected, and I think the country (and I) would usually prefer to pick one direction or the other (though of course individual Elector-candidates could run on platforms of advocating for relatively moderate presidential candidates), but it could be interesting.
You may have noticed the conspicuous lack of discussion of how a Vice President is chosen (and/or that the language about the Vice President in Hamilton's amendment has been crossed out). Since President is by far the more powerful office, I figure let's just leave the Vice President out of the electoral system. My preference might be to have the President-elect choose three candidates for Vice President and let Congress pick one. Or it could be identical to the system for if a Vice President dies: the President gets to pick anyone to be Vice President as long as a majority in the House and Senate OK it. I'm not picky on the details--the point is that no one really cares about the Vice President so let's not involve them in the presidential election process. 
Admittedly, this reinvigorated electoral college would be significantly less democratic than even the current system. The Framers were fine with that, of course--most of them didn't trust the unwashed masses much at all--but I think our modern sensibilities have changed to be more skeptical of the "elites" and to want more power directly in the hands of the people. (See: the Seventeenth Amendment.) However, there is a pretty strong counter-argument to this populist point, and its name is "Donald Trump." If his election can't convince us to go back towards the way the electoral college was originally meant to work, nothing can. You know there is no way that independent smart people would have ever allowed him to get anywhere near power or responsibility.
The other main downside to this idea that I can see is that we would lose the public vetting process that is the current two-year presidential campaign. Right now, the media (and rival candidates!) have months and months to delve deeply into each candidate's background looking for scandals or things that might reflect poorly on their character. I tend to think these investigations usually produce more heat than light, though. They mostly end up discovering scandals/liabilities that have little bearing on actual fitness for office (albeit which inordinately exercise less informed voters ), so in the end I'm not sure we'd lose much.
Conclusion (Yes, this post is finally over.)
So I think those are the two best options for fixing the craptastic mess that is our current method of choosing a President. Direct voting has the benefit of simplicity and increased democracy, while a reinvigorated electoral college might appeal to those wishing to get closer to the Framers' intent and keeping out insane populists.
What do you think? Of course neither will ever happen, but don't let that influence you! Do you want to reform our current presidential election system? If not, why do you suck? And if you do want reform, which of these options would you prefer--or do you have your own ideas?
 Wikipedia has a great overview of this whole process, of course, if you want to do a deep(er) dive.
 Since a state's total number of Electors is the sum of the number of their representatives and their two senators, each presidential voting district will be somewhat smaller than the congressional voting districts, of which there is one for each representative for the State in the House. For convenience sake, I might be fine with something like having the state legislature pick the two Electors that correspond to the state's senators, or having an at-large election for their seats, so that congressional districts can be used. I'm not picky on the details (a theme I will come back to repeatedly in this post).
 For completeness sake, I would also add on a final section giving Congress power to legislate to enforce the amendment's provisions, which for boring constitutional reasons is important. This might allow Congress to, for example, get more specific in legislation about what it means to "publicly express the desire to be President" or for an Elector to "declare who they plan to vote for" and adjust those definitions--and the penalties for violating the amendment--as needed.
 A few things to explain in this proposed language: I didn't want to let the parties decide beforehand who their candidates would be by nominating a person even if they hadn't campaigned for the position, so if a party tries that then their person is out. And I had to specify "by their political party" so that the Democrats couldn't take, say, Jeb! Bush out of the running by "nominating" him :) Finally, I figure five years is a long enough time to prevent someone from "pre-campaigning," but picky on this point I am not.
 Douglas Adams, The Restaurant at the End of the Universe. (source). NB: a lot of sources on the internet, including Goodreads, have "those people who must want to rule people"--I think from context, and a few Google Books sources, that "most" is the better reading.
 The website I'm pulling this from informs me that when this amendment was introduced, the word "Congress" here was replaced by "State Legislatures." I'm not picky on who draws these lines (see note  above).
 I particularly like Hamilton's language about how the districts "shall be formed, as nearly as may be," to avoid gerrymandering--maybe we could add that to the Constitution regarding congressional maps while we're at this?
 Devin McCarthy (Ph.D. Polysci, Duke), "How the Electoral College Became Winner-Take-All".
 I assume that since these are just district-wide elections, Elector-candidates won't start running two years ahead of time like presidential candidates do these days. But it might not hurt to add something to this hypothetical constitutional amendment that prohibits campaigning more than, say, 6 months ahead of the election. That huge reduction on election fatigue alone would almost guarantee that this thing passes! :)
 I do worry about candidates lobbying Electors at this point, though. It would probably be good to prohibit Electors and their immediate family from being allowed to serve in the new administration so that candidates can't bribe them with cabinet positions etc. It might even be wise to prevent candidates from talking with Electors at all during this period (though I'm not sure how well that could be enforced with surrogates etc.).
 Yeah, this would require yet another section in the amendment to remove the constitutional language about how the Vice President is currently chosen and laying out the new process, but that is beyond the scope of this blog post . . . and also kinda boring :)
 I'm thinking of things like Hillary Clinton's email scandal--or even less substantively, the baseless attacks levied against her over the tragedy in Benghazi (not to mention pizzagate!).