Coming from a family who on both sides hail from New York, I can tell you that it's going to be difficult to find an affordable apartment in the NYC metro area, especially on a month to month basis. The rents are insane and if you are talking about renting an apartment for 3 people which would be a 2-3 bedroom apartment, you're talking about 2,500 to 3,000 dollars a month--and that is on the more AFFORDABLE side! You have to be willing to get a 1 bedroom apartment or even a studio with your friends and simply use cots/air mattresses in order to find a place. The studios in NYC alone can be as high as $2000+ every month. What you do have going for you are roommates who can split the rent.
As for locations, Manhattan is as I said, insanely expensive. However the opportunities get better the further uptown you go. Try Harlem, Inwood, Washington Heights, Queens, and the Bronx.
Links: (excerpt from Inwood page) "The combination of less expensive housing, extensive wild parks and access to the water has also attracted a number of artists, students and musicians to the neighborhood.
Inwood appeals to many who seek lower housing costs and, in places, a more serene setting, without actually leaving Manhattan and its subway connections"
http://en.wikipedia.org/wiki/Inwood,_Man…
Washington Heights:
http://en.wikipedia.org/wiki/Washington_…
Harlem:
http://en.wikipedia.org/wiki/Harlem
Queens:
http://en.wikipedia.org/wiki/Queens
For a more affordable apartment, I especially recommend Queens. My mother's family is from there and many of the areas are still close to Manhattan (by subway or bus, it can be short as 30 minutes-45 minutes away, other areas it can be an hour), and you have an abundance of activities, great restaurants, museums, parks, culture, nightclubs, etc.
Great areas in Queens which are very close to the city are Elmhurst and Springfield Gardens where I frequently visited family:
Elmhurst:
http://en.wikipedia.org/wiki/Elmhurst,_Q…
Springfield Gardens:
http://en.wikipedia.org/wiki/Springfield…
There are of course many more areas in Queens that might interest you.
Some links you might find useful are Urban Living.net
http://unfurnished.new-york-apartment.co…
http://www.rent.com/