Here is my list of activities to do in NYC since I have no where else to share it:
Cheap & Fun:
Central Park
The MET
Natural History Museum
Restaurants (my personal favorites, first things that come to mind):
Le Relais de Venise L'Entrecôte Midtown East (they only serve steak and frites, that's it. like $35 for a full course including salad, two servings of steak, and two servings of fries.)
Minetta Tavern (good burger)
Pastis (great burger)
Cafe Reggio (nice cafe, open late)
Lucien (French food, decent spot, trendy though so annoying people go to eat there)
Raku (the one on MacDougal)
uluh (good Sichuan spot)
Spot Dessert Bar (dessert bar on St. Marks)
Little Poland (beef goulash is great)
Bars (barely go out anymore but these places are fun):
Sake Bar Decibal (underground Japanese bar)
Studio 151 (Japanese jazz bar, $20 cover)
Generally can go out anywhere in the LES or East Village, but those two came to mind as places I enjoy the most.
Would try and avoid midtown / Times Square area, but understand people like to visit just to see it. Personally biased to the Upper East Side, but I'm boring. Enjoy and be safe!