After finally cutting myself off of Twitter, my main source for news has been the Wikipedia current events portal:

https://en.wikipedia.org/wiki/Portal:Current_events

It's as close as can possibly be to unbiased, only lists significant events, and it covers world news rather than only focusing on America.

My secondary sources are reddit's /r/OutOfTheLoop, and the chyrons I see whenever I happen to walk by a TV that is playing the news (I learned of the Trump impeachment this way). If something doesn't come up in one of these three sources, it's probably not important enough to be worth the stress/anxiety that following the issue would cause.