I just came back from NYC about 10 days ago. There are lots of great restaurants. If you are looking for a good steakhouse, most people say go to Peter Lugers...although thats in Brooklyn. I really like Del Friscos, but its a bit loud and crowded so make a reservation. Even my wife who is not really into steak, thinks its one of the best steakhouses we've been to in the USA...and she actually gets the beef there instead of seafood (like she does at some other steakhouses).
I have not been to Per Se. But I recently saw a friend who travels extensively and is a true "foodie." He was saying in his opinion Per Se is the best fine dining experience in the US. He said its pricey and not something he would do frequently, but said its definitely something to experience. He also liked it more than its sister restaurant, French Laundry.
If you like old standards/traditional French Restaurants there is Le Perigord, my wife likes it more than I do. She usually likes to go there for lunch. As another poster suggested, La Grenouille, is a beautiful restaurant, and the like frogs legs are an excellent classic dish. The rest of the food was good but not very memorable. Although, I must admit, I would not return. We went there about 2 years ago, and granted we were probably the youngest people there (I was in my late 30s), and we found the wait staff was snotty, and the service rather inattentive and in some cases it seemed they practically ignored us. Perhaps, they felt my wife and I were too young to enjoy their restaurant and so they could hardly be bothered with us. We've been to many really nice restaurants over the years and never been treated so poorly. But maybe your experience will be better.
Another recommendation is Daniel, which my Wife and I both enjoyed it a lot. The restaurant is beautiful, the food is excellent, the wait staff was quite friendly and the service was seamless. It was significantly more expensive than La Grenouille but I liked it a lot more.
For burgers, my NYC friends love Jackson Hole. I've been there once and it was very good, but I haven't been there in a long time.
I believe Patsy's is still one of the more popular/famous places for NY style pizza.
For a nice meal thats not terribly expensive, try the Armani Ristorante just off 5th Avenue. Its modern Italian/Mediterranean. Inside is sleek and modern, the food was very good, and the service was excellent. We went for lunch and they also have a 3 course prix fixe meal, which was only $35 or $40. My wife ordered a prix fixe I ordered a la carte, and we both thought it was a very nice reasonably priced meal. The food was much better than the Armani Cafe I had been to in Boston years ago.
For a snack or dessert, I'd recommend Crumbs cupcake shop...their cupcakes are addictive. Its far better than Magnolias (which is famous for cupcakes but I thought the cake part was as dry as sand). There are also some other nice dessert shops but Crumbs is a favorite snack destination for me.
Good luck and have fun.