I'm sure I've missed many good sources, particularly those written in languages other than English. Unfortunately, at least for the time being, I don't have too many choices.
The New York Times - America's paper of note. All the news that the elite will read, including many American policymakers. Served as the (somewhat) liberal apologists for the Iraq war, thus losing my respect. The op-ed page is usually worth at least a glance, particularly Paul Krugman or Maureen Dowd.
The Washington Post - Longtime second to the NY Times in reputation. The Post still has a number of good reporters, especially dealing with the ins and outs of Congress and the government (which they are right next to). Their coverage of the Middle East is better than the Times, but equally 'mainstream' (pro-warish).
The Los Angeles Times - Despite its reputation as a Republican shill in the not-so-distant past, the LA Times is fast becoming another American paper 'of record'. Their international reporting is uneven, but often good. They also cover my home state, which is something of a plus.
The London Guardian - In terms of good reading, the Guardian's my favorite. They cover international affairs far more completely than American papers, especially the European continent. While many accuse them of being less objective than American papers, I simple think they are more overt with their biases. Also, their website's design is far superior to most.
The Independent - I have a soft spot for the Indy because they're the folks who host Robert Fisk. That said, unless you want to pay, you can't read most of their commentators, which is a shame. If you're creative (and use google) you can usually find their better stories reprinted elsewhere though. Their regular news section isn't quite as palatable as The Guardian, but it's pretty good.
The Sydney Morning Herald - What's best about the Herald is it's Australian, which means you get events filtered through the southern hemisphere. They of course also give one a nice sense of how things are going in 'the land down under.'
Ha'aretz - If you want to know what's going on in Israel, Ha'aretz is where to go. Bear in mind that they are the paper most associated with Israel's liberal elite, but their's is an interesting perspective nonetheless. Their op-ed page is excellent, with particular kudos going to Amira Hass and Gideon Levy.
The Jerusalem Post - I don't like the Post. At all. That said, their regular news reporting is decent. While their op-ed page is somewhat right-wing for my tastes (in general), it gives one a very clear idea of the other side of Israeli politics; something one doesn't see nearly as much in Ha'aretz.
Al-Ahram Weekly - The English edition of Al-Ahram is certainly unlike all the above papers (for starters, it's a weekly). I tend to take what they say with a rather large grain of salt, but they are very much worth reading, especially for their alternative perspective on the Middle East. They discuss things which none of the above papers will. Of course, they're state-run, which does give them something of a slant as well.