Better candidates and better voters
The other proposals presented here deal with refining the machinery of democracy, the rules and systems that constrain its function, but it is, of course, also possible to make better use of the system that we have.
One simple way to fight partisanship and extremism is to vote in the opposition's primaries.
Elections in general tend to favor candidates that appeal to voters with extreme views because such people vote more reliably. This tendency toward extremism is exaggerated in party primary elections by the fact that voting is generally restricted to registered members of the party, which are themselves not reflective of the population as a whole. The result is that general elections in the United States tend to be competitions between two candidates that have more extreme views even than the minorities of the the population that selected them in the primary elections.
This situation would be remedied somewhat if we could remix the membership of the political parties, which we can all contribute to by joining whichever of the two major parties we like least. If a large number of Americans switched parties and voted in their new party's primary election for whichever candidate they liked best, the candidates nominated for the general election would immediately improve.
The disadvantages of this strategy are that it prevents you from helping to select the nominee of your preferred party and that by helping the opposition to select a better candidate you increase the odds that their candidate will win in the general election. The former is certainly a negative, but if you must trust one of the two major political parties to pick a candidate without your input, it seems reasonable to trust the one you like better. The latter is a necessary cost of improving the quality of the opposition candidates, but one that we feel is worth it for an improved democracy. In a functioning democracy, the outcome of most elections is uncertain, so a bad opposition candidate does not guarantee that your preferred candidate will win. In a great democracy, most elections are between two capable, honest, and brave candidates, so a win by the opposition is not a disaster.
Finally, in working to improve our democracy it is important not to overlook ourselves. If we regard democracy as a machine then the fuel upon which it runs, its motivating force, is the passions and actions of the citizenry. Fundamentally, our politics is divisive because we are divided, and while we can and should refine the machinery of democracy to function better with a divided populace and to produce less division, there is no substitute for consensus and goodwill amongst the people. We can cultivate that in ourselves by engaging empathetically with other viewpoints, by speaking and listening to people with whom we disagree, and by reading or watching news from reputable sources whose political leanings differ from our own. In doing so, we may not be convinced by those whom we disagree with, but we cannot help but acknowledge their common humanity.