I've had two difficult conversations since the election has passed with individuals who voted for Trump. Most recently it was with a friend who was bothered by those protesting Trump's victory. I actually shared more of his perspective than he might have realized. Certainly chatting with this friend was easier than the other conversation where my barber was shocked that I was such an elitist that I'd vote for Hillary.
Back to the more recent talk, though. Democracy is hard work. And for those who supported the losing side in an election, there is hard work to do when the election is done. Namely, it's important that those citizens . . .
follow the news (which doesn't mean commenting on every story: it's important to be conversant, not controversial)
support professional journalism (actually subscribing or otherwise paying for good news)
vote in every election, even odd-year primaries; every vote matters
keep humble in knowing that no party or faction owns a monopoly on truth
These are good things to do, too, if one is on the winning side.
By the way, if one keeps their eyes open right now, there's some very useful autopsy results from the election. The results of this election revealed some blind spots in the Democrats' strategy and tactics. They (and the Republicans) were guilty of missing some long-run big-picture trends that matter, and matter a lot. Perhaps the best one I saw today is this one from the New York Times.