Yay for New York! Here is my advice.
1. For a cheap place to stay, check out Hostels - there are two youth hostels here in NYC (I think). Cheaper than a hotel, and a good way to meet some people to bum around with while here.
2. If you want to see theater, forget Broadway! Go see some Off-Broadway theater, too - its costs about 1/2 as much and is often more risk-taking, entertaining and easier to get into. Specific reccomendations: "De La Guarda" at the Darryl Roth Theater, The Public Theater (two shows running right now, one musical, one straight), "Avenue Q" at The Vinyard Theater, "The Exonerated" at the Bleeker Street Theater, "Our Lady of 121st St" at The Union Square Theater. Take it from me - I'm a stagehand when I'm not wearing my pre-med student hat! I've worked on or know people who worked on a lot of these shows and I've heard they are all good, and there's dozens more.
3. Your first stop on arrival should be at any newstand to buy a copy of Time Out New York Magazine. This is a MUST! It'll cost you three bucks, and it will be your best money spent here - it lists almost eveything to do including current museum exhibits, walking tours, music acts, film, theater, sports, dance, and clubbing; including answers to questions like where to buy tickets and where the TKTS booth is and what subways are running on wierd schedules for the weekend and stuff. (Note: many museuems are listed in the 'around town' section, not the art section). The listings run from Thursday to Thursday so you may have to pick up two copies if you are staying for a different time interval, but it will tell you so many things to do that you will not be bored! Whenever I have traveled to new cities that have Time Out magazine I always buy it, and I end up doing stuff I never would have known about otherwise.
Have fun while you are here, and take care not to get your pocket picked!
-rebaj