Going to Newark and Queens might be a bit out of the way if you're trying to stay on the island. If not, then all of those spots mentioned are rock solid. I can give you some Brooklyn recommendations if you'd like.
My favorite Italian spot in Manhattan: Pesce Pasta in the West Village. Delicious, affordable, and great atmosphere. Blind Tiger is a great bar around there if you like beer and there is always some great off-Broadway Theater at Barrow Street Theater and films at IFC Center (depending on when you go, there is always something great on weekends at midnight).
Indian food: Lexington Ave between 28th and 26th streets is as good as it gets. Avoid Curry in a Hurry, but the other spots are great. Nirvana may be the best of the bunch.
Pizza - Certainly up for debate, but arguably the thing that NYC does best. My top-3 are in Brooklyn (Di Fara in Midwood, Lucali in Carroll Garden, and Totonno's in Coney Island), but John's (West Village) and Patsy's (East Harlem) are pretty great. Avoid Grimaldis and Lombardi's. Those places are tourist traps. Also, if you're a fan of Louie, then check out Ben's (also in the West Village).
My favorite concentration of restaurants, if it wasn't obvious already, are clustered west of 6th Avenue below 14th street and above Houston. Some of the spots are expensive (Po, Market Table, etc.), but you can't go wrong with any of them.
Another thing I haven't seen on here are Delis. Carnegie and Katz's are "experiences" - really expensive, but good enough, especially if you're not in NY that often, but Mile End in Downtown Brooklyn is, by far, my favorite.
Anyway, I love talking (and arguing) about food in NYC. It's the thing I probably miss most about the city, so let me know how far you're willing to go and what kind of food you're into and I can definitely give you some recs.
The one thing that I can recommend off-the-grid in terms of tourist things to do yourself a favor and walk over the Brooklyn Bridge at sunset. Some others, especially if you're not in NYC often - the High Line and a ride on the Staten Island Ferry to see the Statue of Liberty up close (the best part is that it's free).