From EJP Photo. Far better advice than “Get your new man to buy you something from Tiffany’s”.

To answer your question, I’d stay away from the tourist traps (Empire State Building, Statue of Liberty, etc). They’ll suck up your time, they’re really not much to see up close, and you’ll only be hanging out with other tourists. I’d stick to the stuff you don’t need a ticket for - Times Square, Central Park, Washington Square Park*, walk the Brooklyn Bridge. What really defines NYC is the unique culture and diversity, and those are the kinds of places you’ll find the natives. The only exception I’d make to that is for the museums, of which there are many well worth experiencing. But eat the pizza (I recommend Grimaldi’s, under the Brooklyn Bridge on the Brooklyn side), stop in bars, and check out anything that catches your eye.

Have fun!