I've lived here for almost three years now. I'll second a few items:
Museum: MoMA.
By far my favorite museum in New York. The permanent collection is seriously like walking through an Art History 101 textbook. It includes important works by Monet, Matisse, Cezanne, Van Gogh, Picasso, Dali, Mondrian, Warhol, and much more. The new space is also spectacular and many of the temporary exhibitions are great. Also, the design collection is cool and unusual.
Also, if you're coming soon, the Guggenheim has a really cool exhibition now through May 28 by
Cai Guo-Qiang, the centerpiece of which is nine Chevy Metros (seven of which are suspended from the atrium ceiling) in a progression that simulates a car bombing.
Restaurant: Otto
This is Mario Batali's "low-end" restaurant, which means you'll probably get out at around $40 per person. The menu is just antipasti, salads, thin-crust pizzas, pastas and gelati. Go with a group, get a lot of things to share (I especially recommend the Pizza Vongole with fresh, in-shell clams; the pasta with hazelnuts and butternut squash; and the shrimp with chiles). The Olive Oil Copetta dessert (olive oil gelato with salt and a seasonal accompaniment like blood oranges) is unique and super-delicious. They also have a wide selection of high-quality wines by the quartino (1/3 bottle).
Bar: Pegu Club
This place's focus is innovative mixology, so you'll see lots of unusual drinks like an Earl Gray Martini. Unfortunately, I find the menu to be a mixed bag: I've had some drinks that were unexpected and outstanding, and others that I found unpleasant (usually because they were too bitter). It is never boring, however. I find the champagne cocktails are the best bet. The atmosphere is super-chill and plush, and you can usually get a seat.
Transport: The Subway
If you're not from New York, definitely try to explore by subway. The New York subway is unlike any other system I've seen (and in fairness, I haven't been to Tokyo) in terms of the sheer volume of people moved. Day and night, the system is busy. I've gotten used to commuting on it every day, but I still think it's a sight to behold for outsiders. Also, it's an outstanding deal.
However, if it's 2 AM and you're drunk, definitely treat yourself to a cab.