If you want a very UNIQUE New York City experience, be sure to have dinner at Lucky Cheng's. (They have a website, I think it's simply www.luckychengs.com, but I'm not positive.) If you visit the site, don't let it scare you--all the employees are incredibly friendly, and the whole experience is a hilarious riot for everyone involved!
Any first time visit needs to involve a trip to the Statue of Liberty. Hint--order your tickets online ahead of time, and be sure to get the time-stamped pass (limited quantities) ahead of time that allow you to go up inside the base of the statue. (Since 9/11, you can no longer go to the crown--but the view from the balcony at the top of the base is pretty amazing, and you can see up into the inside of the statue.
If you want to go up one of the skyscrapers, be first in line when they open, or go very late at night--otherwise you're likely to be stuck in long lines. The Empire State Building view is still classic, although I've heard that the view from the top of Rockefeller Center may be even better. (I've not done it myself, but I've heard you get much better views of Central Park and Times Square.) And the late night (dark skies) views are quite neat to see--the lights of Times Square from that height really stand out.
Just a bit south of Times Square on Broadway is Little Korea, with a lot of really good food. If you're not familiar with Korean food, though, be sure to find a place where the waiter/waitress can speak decent English, to teach you a bit about what you're eating. Otherwise you'll be lost. Pick a place that has wood-fired, at the table cooking--they put the coals into a hole in the middle of your table, and cook the food right there. (At some, they cook it for you; at others, you do the cooking.)
If you're a fan, check the Late Show w/ David Letterman's website (link is on www.cbs.com), and see if they're filming when you're there. If so, apply for your tickets now, and then do a lot of praying...
