This is something that I don't think is well understood: the most broken component of most news organizations is the editing. News editors tend to be...not smart. They don't have nearly enough subject matter expertise to make intelligent decisions about how to asses factual claims in articles or how to put the news in context. If I want to read about economics, I know for a fact that any econ blogger on the planet knows a hell of a lot more than 98% of newspaper editors. What's far worse is how news organizations present politics; they behave as if political science doesn't exist. Everything is explained in terms of fictional narratives that often contradict polling while ignoring fundamentals. Reading news articles about politics makes you more ignorant, not less.
For example, I read Lawyers Guns and Money ( lawyersgunsmoneyblog.com/ ) and when, say, a war in Georgia breaks out or a South Korean warship sinks, I know about because the LGM guys write posts with links to news sources. But since some of them are security studies professors who specialize in international relations and global security, I get the added context to make sense of it.
So pick some subjects that interest you, find some subject matter experts with blogs and toss them into an RSS reader. The result will be better than any newspaper and you'll still get notified when an "important" article gets published in the NYT or WAPO or LAT or WSJ or anywhere else.