Just for yourself? Shit... that's quite a lot, you can get a studio almost anywhere for that price and a 1BR should be easy to find as well if that's what you prefer. Go downtown for hip and trendy or LES/Hell's Kitchen for a little more... sophistication I guess.
If you want a 2BR you'll mostly be relegated to Uptown Manhattan, Brooklyn, and Queens. Still, very nice areas for that price. If you want a 3BR for $2000 you need to start going a little further Uptown Manhattan... North Harlem, Southern Washington Heights, at this point I would avoid East Harlem, Inwood, and Northern Washington Heights - it'll start getting a bit shady. You can also go into East Brooklyn, but the further you go East the crappier it gets, for 2000 you can get a nice 3BR though in Bushwick just do your best to avoid Bedford Stuyvesant, East Bushwick, East New York, and the shitty parts of Crown Heights at that price range.