There is a great show called Stomp that plays downtown, it is a must-see high energy percussive dance show. You'll love it.
http://www.stomponline.com/
If you walk a little west of the theater there is a cool neighborhood called Alphabet City (Avenue A, Avenue B, etc) that is an upandcoming trendy area for restaurants, wine bars, and clubs.
You can't come to NY and not see either the Modern Museum of Art (MoMA), or the Metropolitan Museum of Art.
The Yankees are in the post season so if you can get tickets that may be fun. Hockey season has just started so you may be able to get tickets for the Rangers at Madison Square Garden.
Here's a website of all the different places in ny, it could be a starting point.
http://www.thingstodo.com/states/NY/nyc.htm
On often overlooked area is the meatpacking district, very cool restaurants and pubs.
If you like Asian food then head down to 6th street, it's like a Little Japan. Or go further down to Canal Street and hang out in the exotic streets of Chinatown.... Little Italy is stationed in the middle of China town on Mulberry Street so you can have dumplings and cannolis all in the same day.
The true beauty of NY really begins about an hour north of the city. NY State is really a beautiful New England staple, autumn being the best time. If you have access to a car while here I'd be happy to give you more info on hiking and other activities to do outside the city.
Whatever you do stay away from Times Square. Overpriced and kitchy, it's an area that New Yorkers stay away from.
Also, check out Time Out NY, it's a magazine but it's also online and it's the best source for all events and activities going on in NY from music to theater to art, to dining, to so much more.