If you have not been, don't avoid Times Square and the WTC. I understand that residents view that as touristy. Guess what. You're a tourist.
You have to make your own plans, but here are some things I have enjoyed.
The Ground Zero Museum (
http://www.groundzeromuseumworkshop.com/) I can't tell you in words how powerful this place is. If you go, only buy your tickets from his website, not through any of the brokers.
I was unwilling to wait for the ferry to go to the Statue of Liberty, but there are water taxis out there that provide a great view and a little bit of a tour of the area.
Chinatown - knockoffs galore. You can get all your Christmas shopping done for about $50.
As badly as I hate to admit it, Yankee Stadium is awesome. Haven't been to Citi but odds are one of them will be in town when you are there.
I will never go back to NYC without a Sunday being involved so I can go worship at Brooklyn Tabernacle. Been twice and it's amazing.
Hopefully you're going to take in a show on Broadway. Always a good time.
You can find tons of good places to eat. Use TripAdvisor for some guidance and OpenTable to help make reservations. You accumulate points with Open Table and they send you gift certificates too. In fact, you can go with 500 point restaurants and have a gift certificate in no time.
If you like breakfast, go here, but go early. It only sits about 40.
http://clintonstreetbaking.com/
I accidentally ended up at Katz's Deli one day. If you go there, no matter what you do, DON'T LOSE YOUR TICKET.