If you get breaking-news notifications on your phone, you'll often find you disagree with what Apple about what constitutes news. Instead, you can turn use news aggregation app Nuzzel, which can notify you when a certain number of your Twitter friends link to the same URL, no matter the source.
You can also set a daily maximum and get a daily email digest. When you open a story, you'll see what your friends tweeted about it. It's like an aggregated comment section that doesn't suck.
Nuzzel's social-based approach works great when mainstream coverage misses important context, when news comes from an unusual source, or when the day's conversation revolves around something not considered "news" by traditional publications, like a Facebook feud or an embarrassing op-ed. Instead of "training" Google or Apple what stories to show you, you just connect your Twitter account.
If you follow people with similar values and interests, you'll have a ready-made feed of the news you actually care about.