I have to agree with the other posters, New York is an excellent choice for a first US vacation.
Vegas isn't representative of the best of America and is limited.
New York offers such varies experiences. You are interested in shows, New York has some of the best! If you can try to see a couple broadway shows (maybe off broadway) one high drama and one stomping old school musical. And the museums and the art galleries! There are also so many varied neighborhoods in Manhattan alone. You should cheek out Brooklyn as well, a neighborhood such as Williamsburgh.
The food is not just a myth, the food in New York is unbelievable. I would travel from England for a proper bagel alone! Aside from the bagels and pizza you can get top notch ethnic cuisines, Chinese, Japanese, Greek and Indian stick out to me.
You are also two hours from the Hamptons...perfect for drinking with the girls. And you are about 3 hours from the beautiful Catskill Mountains of upstate New York. You may not get upstate as you probably have about a week (if more, awesome for you) and the city will keep you busy. Upstate is beautiful through.
As others suggested, do your research before you hit New York. It is so big, if you just wander about you may miss the best of it. And yes it can be super expensive but with a bit of planning can be manageable. Look up reasonably priced neighborhood restaurants, you'll enjoy it more than tourist traps. Timeout NY is an excellent resource. Do go see time square, it's a total trip...just don't waste your dough there
Good luck
It sounds like you have fun plans to make!
http://http://www.nyc.com/restaurants/
http://http://www.timeout.com/newyork