Things will be easier with a roommate, so think about what matters to you.
Do you want to be near the hip and trendy people, cafes and nightspots? That’s the East Village and Lower East Side in Manhattan or Williamsburg and Greenpoint in Brooklyn, or Astoria and Long Island City in Queens.
Do you want to be near the water? That’s The Financial District and Battery Park City in Manhattan, or parts of Queens, like Far Rockaway.
You could live in Morningside Heights, near Columbia University. It’s pretty cheap up there and there’s lots of young people around, thought there’s also more families. Lots of people also live in Inwood, though the train ride will be a little long at night. Apartments in Inwood are pretty big compared to many parts of the city.
You could live in Harlem, or the Upper West Side. If you want a similar neighborhoods that are a little quieter, there’s Fort Greene and Clinton Hill or Boerum Hill, Cobble Hill and Carroll Gardens in Brooklyn.
The Upper East Side is OK, but the transport options there aren’t as good, and the apartments are more expensive for less space.
I don’t know much abut the Bronx, to be honest, though I’m sure there’s great neighborhoods there, too. And I think St. George on Staten Island is nice, and it’s right by the (free!) ferry, but that’s the only neighborhood I know there.