This is a great point and pretty much mirrors my own views. I read the NYT but I'm under no pretense it's "neutral" or "fair and balanced". It is a highly partisan, albeit factual, news source that is selectively edited and clearly pushes a very specific worldview, through which every story is filtered. This worldview includes beliefs like: unions are always good, capitalism is at best suspect and makes a few rich, at worse a primary driver of inequality, all inequality is bad, middle-class wage stagnation is a huge problem, all government spending is good, regulation is the answer to any problem and any who oppose it are misguided/stupid, all law enforcement is racist/generally suspect, and my personal favorite, that some cabal of powerful people meet and regularly conspire to screw over ordinary "middle class Americans" (my wife and I call these hypothetical people "mustache twirlers", generally evil white men with white Persian lap cats and snifters of expensive cognac).

As a practical matter, I think the only thing a person can do is consciously seek out both sides of the story. Read a credible conservative daily (WSJ, Chicago Tribune) and a left-leaning Sunday paper (WaPo/NYT)/news magazine. Get both sides of the issue and see which you find more persuasive.