Being originally from North Jersey and in the New York media market I usually read: The New York Times, The Star Ledger, and The New York Post (a habit I picked up from my father).
Currently residing in New England I usually read the Boston Globe for local news.
Other places I get great journalism from: Slate, The Atlantic, Sports Illustrated (sometimes), The New Yorker.
These are the ones I check regularly, but I'm open to new suggestions, hence the question.
Honestly it's hard to say nowadays. The short answer is via a combination of aggregators. Through those, I end up getting a lot through the NYT, Atlantic, BostonGlobe, Wall Street Journal, BBC, and any number of smaller sites, blogs, and social network contributors too numerous to even begin to mention. So I soak it in passively that way, but should I be interested in a topic in particular, then I will do google searches and social network scans to mine links to reporting that is more in depth. Outside of the aggregator and social network curation, the only news direct from the source I generally tune into is NPR in the car. I don't think there is a single new organization in the world that is trustworthy or dependable as a single source for reporting on every important topic though, especially when it comes to investigative reporting that lives up to the most romantic idealizations of the "Fourth Estate." I also find that I increasingly seek out and turn to specific reporters via social networks, and respect them as their own brand, separate from the organizations that pay their salaries.
Hubski, Google News, Popurls (oddly enough), ProPublica, and NPR. Using Hubski, Google News, and Popurls means that I am usually getting it from a variety of sources, but I make a special effort to avoid CNN, MSNBC, and FoxNews. If I see that they have a headline that interests me, I'll actually Google News search for another source, like the NYTimes, Washington Post, BBC, LA times, etc.
Al Jazeera and the BBC are the two sources I trust the most. Beyond that, aggregations and a newspaper.
Hubski, NPR, the Economist, the Atlantic, Scientific American
Twitter, reddit, Google+ BBC and the Economist I find that most stories have economic conections - I try to cross reference news stories usign the BBC Reutors and then i try to research the actual thing they are discussing to see if they are just making it up or manipulating data to make it fit a story.
Recently, I've started using Digg again. It has pretty good and interesting news. It's not for specific interests but usually includes a couple of things you want to click on.
As many people are saying in this thread, I get most of my news from aggregators. However, one site that I don't think many people know about is reading.am. It lets you share exactly what you're reading, and it lets other people see this. I find that the front page usually contains some good reads. It's a really simple website, and I love it. It's become almost reflex now to just click the bookmarklet to share what I'm reading.