Have a great time. My first trip to NYC was the summer after my senior year and I was so excited that I threw up on the plane.

Hope you don't do that! I have been countless times since, but that trip holds such a special place in my heart. I saw four shows and counted myself lucky. You should truly count yourself lucky for all the ones you'll be seeing, plus the TONYS! My guess is that you are going to enjoy all of them. There is nothing like your first Broadway experience and you'll be blown away by the talent, whether or not you enjoy the story. Keep a journal of your thoughts of the shows, NYC in general, where you eat, who you see, etc. You'll love to read it in ten years.
I agree with the person who suggested the NBC tour. I think you and your friends will think it is interesting. It sounds like you'll be pretty busy, but if you get a chance, go down to the Village and wander around. As much as you'll love Times Square and all the Broadway stuff, you should try to see a little more of the city if possible. If not this time, some other time.
HAVE FUN!!!!!