As mentioned above, make sure you negotiate hard! I’ve seen friends get first and last month free, minimal deposit and less stringent on contracts.
I’ve got a buddy that lives in the Upper West Side, on 80th between Amsterdam and Broadway. He’s only paying 1,000/month after him and a few other guys split their rent four ways. His place is tiny, but I’m sure that will be the case just about everywhere. Anyway, he really loves it, and from my tourist point of view the neighborhood seems great. All that being said, you should probably follow the craigslist advice and try to find a place to split with roommates to make things much more affordable.
I would recommend UES in 70s or 80s – plenty to do at night… Plus you can find a decent studio for $1400-1500 (even 1350 if you look harder)… Make sure it’s rent stabilized apartment though…
I would generally avoid Chelsea for reasons listed above, you get no space for ridiculous rents. Check out east village as a starting point - you’ll get more space and I think its a pretty awesome area. Check out places around union square. I live in hell’s kitchen and its not bad rentwise but no great nightlife and its close proximity to Broadway gets really annoying when you have to fight through tourists all day long. I would also avoid Fin. District as it shuts down after 7pm.