Figure out your budget and the neighborhood you want to live in and then look for ads that are similar to what you're looking for. There are deals to be found, but they are very few and far between. These are all easily accessible by subway and bus, and most have a number of fun things to do in the day or night. Once you've found a few ads you like, call immediately and if the apartment or room is available, and you feel good about the person on the other end of the line, set up a time to see the space. If you like the place and the person or persons you'll be living with, or the landlord you'll be renting from, find a lease or sublet agreement that you feel comfortable with.
Many places ask for a large amount of money up front.
If you're worried about handing over your life savings in exchange for a set of keys, call the landlord or owner and make sure your agreement is above board. Also, don't get so excited about finding that apartment that you sign on to spending more money on rent than you can actually afford. When you're living on tap water and ramen that next month, you will regret it. New York City's public transportation system will take you everywhere, from Coney Island to the Bronx Zoo and back to the West Village for a night of comedy, jazz, and everything else you might expect out of New York nightlife. There are restaurants everywhere, and you certainly won't have to travel an hour for good Indian food anymore.