Hi there! I'm new around these parts, but thought I'd jump into this conversation because I read a lot of literary magazines and am generally a huge mouthbreathing dork about it.
I live in a large, extremely artsy college town in Massachusetts, so part of the reason I read so many magazines is that access to them is really really good (even our local Barnes and Nobles stocks some). As others have said, The New Yorker and McSweeney's are really good. (Roddy Doyle is my jam. Always. What a badass.)
Some other good magazines to try (in rough order of availability):
-The Paris Review (thoughtful, litsnob, very highbrow, rarely anything but realist, really excellent interviews and criticism.)
-The Kenyon Review (similar)
-Tin House (tends to be more variable in genre-- they do themed quarterly issues-- although also more variable in quality, in my opinion. They also have good interviews from time to time.)
-The Massachusetts Review (some great, very well-established writers regularly publish here, but also some up-and-comers, which I like)
-The Common (published through Amherst College. Great small magazine, very, very professional, more very respectable litfic names doing their thang.)
Overall, if lit magazines usually aren't your thing, I'd start with Tin House and The Paris Review.