Now, take this with a grain of salt since I haven't lived in New York City since 1991. But hopefully I can at least begin to steer you in the right direction and NYC-based GreekChatters like XO Kathy and Cream can help you out some more.
Manhattan is definitely the most expensive place to live. A quote that comes to mind came from the director of an internship I did in college: "financial guide books will tell you that your rent should take up no more than 25% of your salary. If you read that in Boston, you'll laugh and if you read that in Manhattan, you'll cry!"
I lived on 67th and First when I first got there, in a 3 bedroom apartment with two roommates. I think the total rent was $2,300 a month.
Once I was established in my field, though, I wanted my own place and I ended up in a Brooklyn neighborhood called "Park Slope". I'm not sure if that's near where you'll be teaching, or not, though. It's a pretty quiet neighborhood (or as quiet as living in a city can get) where everything I needed was within walking distance. I had a nice, albeit old, one bedroom on the third floor of a brownstone and I believe I paid $925 a month for it.
Some of my co-workers lived in Brooklyn Heights, which is a similar, middle class sort of neighborhood. I recall that the rents were in the same ballpark as mine.
As for as finding an apartment, I was exceptionally lucky both times. Two friends of mine who had just graduated from Columbia needed a third so I just moved right on in. With my own apartment, a woman I worked with had accepted a job in Atlanta so I sublet it from her.
I know a lot of people use brokers, but they charge a percentage of the total rent on the lease and that can really add up.
Will you be staying on L.I. until you find a place? If so, I'm sure once you get to know people you can get some leads on a good place to live. Seems like a large number of apartments are found through word-of-mouth.
Also, does NYC Teaching Fellows offer any housing assistance?
Sorry this is so fragmented! I'm just tired tonight. Best of luck with your housing search! I really admire the job choice you made!
