It all really depends on your budget, interests, and nostalgia.
For pizza nostalgia you go to Lombardi's in what was once Little Italy. It's coal oven pizza which is my favorite but many people prefer gas or wood. Also, many people will recommend better pizza places, that's why I said for nostalgia. John's on Bleeker is very good for coal but if you prefer gas you can go to Joe's. Joe's is what most people would say is a typical NY slice.
Other nostalgic places are Katz as mentioned above, Nathan's, Junior's, 2nd Ave Deli, etc.
Steak: Peter Luger (also nostalgic but also what most NYers will point to as the best steak house in the city)
Japanese: Yakitori Totto
Michelin Stars: Eleven Madison Park
Chinese: The Bao (be sure to try the Xiao Long Bao)
BBQ: Fette Sau
Culture: Museum of Modern Art, Natural History Museum, Guggenheim, etc, Broadway show, 9/11 memorial, United Nations tour, etc.
Really, the list is endless and really hard to pin down any one thing you need to do. For me, I pick a region of the world for each day and discover my own food tour. But other people wouldnt care about that.