Get the NYC DOT bicycle map (download or free at bike shops). Also note DOT has information on bridge closings. Sometimes if a bridge is closed they have a free van service. Stick to the paths on the map, watch out for pedestrians, and car doors. Good luck.
www.bikeforums.net is a good source of info. See info on avoiding bike theft. Hint: get a strong U lock and take the seat post with you.
Consider buying a folding bicycle in NYC and returning with it. bfold.com, nycewheels.com and others have folding bikes. Strida would be a good bicycle for touring NYC. Dahon make very nice general purpose folding bikes. Folding bikes can usually be brought on the subway and trains. A bit iffy on crowded buses.
New York has some of the world's best urban bike paths. The key is to rent a decent bike. Avoid the rusting junk rented in and around Central Park. Instead go to a good bike store like Bike Renaissance (81 Street and Columbus Avenue, across from the Museum of Natural History) or Toga Bikes on Amsterdam at 65th and rent yourself something nice.
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