I was born in New York, and get back there whenever I can. Seeing the City like a native is different than coming in for the first time. I personally like the sidewalk vendors and food trucks to eat. Many restuarants are geared for tourism and over priced. The sidewalk vendors are more reasonable. I suggest you stay away from the big over priced restaurants. I love to see a show when I am there. Off Broadway is much more reasonable and you do not need reservations a year in advance. Central Park is a place I truly enjoy and. I love the many Museums, as there are many to choose from. When you are there personal choices will make a difference. Seeing the Reflection Pond where the Twin Towers stood is a place I love to visit just to pay my respects for those who lost their lives from 9/11. China Town is great. It has a mix of shopping, restaurants, museums and showcases a different culture.
You asked what I like and my personal opinion is New York is such a melting pot of cultures, it offers so much, but you have to decide what you would like. I love Madison Square Graden and Rockefeller Center especially in the winter time. The Ice Skating rink is wonderful. Central poark for a nice Picnic and a Carriage ride in the evening.
I would stay away from the Chain Restaurants like Olive Graden as you want to taste New York, nit something that you can get anywhere. Carnagi Deli and Katz Deli are nice, but very expensive, and that as a student is what you want to stay away from. This is a list of Deli's that you can choose from: http://www.zagat.com/lists/best-delis-tops-restaurants-0
Shopping, I love Macy's! My Mother worked for them and Macys in NY, is nothing like Macys anywhere else.
You may want to try and get a ticket into Letterman show. It is at the Ed Sullivan Theatre , this link tells how to get tickets: http://gonyc.about.com/od/tvtapings/p/david_letterman.htm
Please let me know if you need more. Once you get to New York, it is a whole new world. Joan