Well the Met is huge, lots of things to see but low on photography. You'll have to go to some of the more modern, more expensive museums. MoMA is good, but I think $20 to get in. Try the Guggenheim, too. For revolving exhibits, pick up a couple of papers the first day you're there, definitely the Times and maybe the News and the Post. They'll let you know what there is to see.
I'm not much into shopping, but you can get cheap "designer" stuff on Canal street. You can usually find little stores that aren't extremely expensive. Try midtown and look for things that barely look like stores. I was in one place that just looked like a shoe store, but then the basement was huge and filled with clothes.
I wasn't 21 last time I was there, but you could always go to the real Coyote Ugly, just to say you went.
Have a great trip! Where are you going in Europe?