Here are some ideas en vrac:
- The Bronx zoo is nice. While you're there, Fordham University is pretty, the Bronx Little Italy is cool, and Calabria Pork Store's heroes are to die for.
- In the same area, if you are into Edgar Allan Poe or the urban history of the Bronx, the Edgar Allan Poe Cottage is interesting.
- Ride on the Roosevelt Island Tramway
- If you like art deco (or have no idea what it is), visit the lobby of the Chrysler Building
- Sample all the types of foods you can't get at home. Some of my favourites are noodles and pork burgers in any Xi'an Famous Foods restaurant; pizza (Suprema on 8th Ave at 30th St is a good one); deep-dish pizza at Emmet's; expensive but sublime Roman food at Lupa; soul food (I tried Country pan fried chicken on 8th Ave and 151st St and liked it a lot, but that was a long while ago); cheese cake
- American Museum of Natural History
- Met Opera: Aida or La Bohème if you don't really know opera or want a classic, Samson et Dalila if you want something different
- Brooklyn Bridge
That's what comes to mind quickly. Whenever I go to NYC, eating and wandering around are my favourite activities. All boroughs are interesting to visit (Staten Island admittedly a bit less than the others). Don't forget Harlem. Don't let people tell you NYC is dangerous, it's not anymore. Don't buy a ticket to Ellis Island until you've taken the Staten Island ferry because you might well find the view you'll get entirely to your satisfaction.
Enjoy!