If you're feeling very touristy, take a Grey Line tour of the city, it's a good way to do the sightseeing in a day. Also trips to the Statue of Liberty and Ellis Island can be done.
If you got youngsters with you, the Liberty Science Center in Jersey City is a great place to go, a hands on science museum, with an IMAX in it.
If you enjoy baseball at all, Yankee Stadium is a must go to, but don't get the bleacher seats, since they restrict your access to the stadium. Shea Stadium is home to the Mets and is out in Queens, and I only suggest it if you really want to see that stadium as well. (Side note, don't root for opposing teams there)
If you feel the need to go to a Broadway show, check out TKTS in Times Square for the cheap-ass tickets.
As far as cuisine is concerned, Grays Pappaya on either 8th St at 6th Ave or 38th St at 8th Ave, it's quite possibly the greatest hot dog you will ever eat, and very good drinks as well (get a juice based drink and not a soda)