I know nowadays people uses Craig's list for apartment searching but I like NYTimes the best.
NYTimes Real Estate Section
http://www.nytimes.com/pages/realestate/
Other resources
Craig's List
http://newyork.craigslist.org/
Village Voice
http://www.villagevoice.com/
For NYTimes, do advance search for Midtown East side and Gramercy/Flatiron.
If you don't mind taking buses or subway, look into East Village, Upper East Side, and Lower East Side.
East Village and Lower East Side are fun area, if you like drinking and eating out. Many students and artists prefer to live in East Village and Lower East side so sometimes disgusting and crummy places could be very expensive. But if you prefer more residential area, go midtown and Upper East Side.
It's been so long since I looked for apt in Manhattan so things could be different from what I remember.
Anyhow, here's my advice and info for you
-Real Estate broker will take 15% of yearly rent for commission.
-Its very rare to find ads posted by apartment owners on NYTimes. Most ads are posted by brokers.
-You will more likely to need a guarantor to co-sign the contract. Find rich parents/uncles/aunts to do that for you.
-Real Estate agents don't show apt more than a month ahead of the time you will need the apartment. (That's how agents pressure people to rush into making decisions and they could keep the rent high.)
-Sometimes, you have to put deposit for 2 months rent.
-Try to look for a rent-stabilized apartment otherwise the rent will go up ridiculous amount every year. (It is almost impossible to find rent-stabilized apt though)
-If you find a place you like, be ready to sign immediately. (Although I recommend you to look around at least a few to see what the current market for apt renting is like these days.)
Finding a good apartment requires determination and luck.
Good luck.