NYC kinda sucks, i stayed with my friend there for 3 days last summer and was bored out of my mind after day 1. the museums and sight seeing stuff is cool and all, but past that, meh.
as far as cool shit to do in America, shooting guns is a given, I've taken Japanese and French exchange students, a kid from Norway, and a couple kids from Canada shooting on separate occasions, to my redneck friends house. he's got the most badass collection of guns, plus dirtbikes, stuff to blow up, beater cars to take off his dirtbike jumps, etc.. and each of them thought it was the coolest shit ever. so find some rednecks and hang out, you'll have fun.
other than that, cant help you with NY