What exactly is there to do in a big city?
Everything!!!! As a born and bred New Yorker, I can tell you that it's impossible to run out of things to do here!
I'm assuming you're into music, so I'd recommend:
The Philharmonic (you can usually get tickets for pretty cheap)
The Metropolitan Opera (very expensive, but worth it if you're into opera)
Jazz clubs (some of the best ones are Smoke, the Blue Note, the Village Vanguard, and the Jazz Standard. Jazz at Lincoln Center is also a great venue for jazz).
The Chamber Music Society
The Miller Theater at Columbia often has some interesting stuff.
A Broadway musical! There are so many good ones playing right now.
Steinway Hall is incredible for any pianist to visit.
Some non-music stuff that I'd still recommend:
Museums! In my experience the best ones are the Metropolitan Museum of Art (which has a choose-what-you-pay policy), the Museum of Modern Art, and the Museum of Natural History. The Frick Collection, the Whitney, and the Rubin Museum are smaller but often have some interesting exhibits. I've heard that the Brooklyn Museum is really great as well (but I don't really go to Brooklyn

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The ballet (again, expensive, but worth it if you're into ballet. The best companies are the ABT and the NYCBT. I personally wouldn't recommend the Nutcracker because it's very cliche and touristy, but to each their own).
Theater - both on and off-broadway.
The Bronx Zoo
There is so much great food in the city and not all of it is as expensive as you might think! Check out TripAdvisor for some restaurant ideas.
Some things I wouldn't necessarily recommend:
The Radio City Christmas Spectacular. Honestly, if you want to see really good music, dance and acting, there are thousands of better options in the city, some of which are cheaper. The Christmas Spectacular is just kind of childish and goofy. If you're really into Christmas, it might be fun. But I think there are better things to do.
Holiday shopping is great in New York, but I'd steer clear of places like Times Square and Herald Square unless you're prepared for gigantic crowds and endless lines.
But in a place like New York, it's really hard to go wrong. If you just wander around some hip neighborhood for a little you'll probably find something amazing. It's great to have plans in advance, but it's also great to just enjoy the city and see what comes your way.