My 1st trip to NYC (New York)

Hello,
Actually I live and work in Moscow, and I will be in NYC for the 1st time in my life, from the 11th until the 16th of May. So I would like to get your advice, tips about very interesting, nice, unusual places, sights, restaurants, clubs etc.
In fact, I know nobody in this city...and I'll travel with a girl, one of my colleagues at work.

I would love to get to know the REAL New York (or at least, some part of it :)), and not only what traditional tourists go to visit...

I would really appreciate your help and tips!
Sincerely,
Eva Lebedeva