What are the "must-do's" in New York City for a first-timer?

I’m making my first trip to NYC in early March (way overdue!), and need some relatively inexpensive ideas of what to do/where to eat/what we must see/reputable tours. We’ll probably be there Wednesday-Sunday, so we have quite a bit of time.

We’re staying with my boyfriend’s godparents – who happen to have an apartment right by Time Square, so we will be in a great location. We’re both 21 year old college students, if that makes a difference in your suggested activities.