Joolz,
The Hilton is on 54th Street and 6th Avenue, so you probably want something close by. You don't say what day of the week you arrive. That would make a difference -- not all restaurants are open on Sunday night, and Friday and Saturday toward the theater district can be crowded pre-theater (i.e. 6:30).
You also don't mention your price range, and the prices can make a huge difference.
If you have the money, Aquavit is wonderful. Great fish, inventive, terrific chef and although they have moved since I was last there, the review indicates that the decor of the new place is as nice as the old (which means very sophisticated, Scandinavian modern... you'll like it.)
Cite is another good option for steak.
If money is no object, Le Bernardin is a fabulous fish restaurant.
For great Italian and a fun atmosphere try Osteria del Circo.
Ruth's Chris is an American steakhouse -- part of a chain, but excellent steak in a dignified atmosphere.
Carnegie Deli doesn't fit your food descriptions -- it's a Jewish delicatessen famous for enormous sandwiches and snappish waiters -- you might get a nice one, but they are famous for being unfriendly. If you like this type of food, you might want to try it some time during your visit -- it's close by and it's famous.
Molyvos is a delightful Greek restaurant owned by a family that takes great pride in their eateries. I have a great meal there every time I go, but like most places in this neighborhood, it's not cheap.
If you are on a tight budget, a good option is Menchanko Tei. This is a Japanese noodle shop -- light on atmoshphere, but very authentic, inexpensive, Japanese noodle soups and other dishes.
If you go a little further afield, to the west 40's (theater district), the recommendations I put in an entry for someone a few weeks ago, would also be good. If you love Indian food and want to go to one of the best restaurants in New York, try Tabla. You'll need a cab down to Madison Square (25th and 5th), but the food is fabulous. It's Indian/Continental fusion and the chef is a genius.
To do your own research, google Zagat and click on New York Restaurants. Check the listings under "west 50's" and you'll see what's available.
Enjoy your visit to New York!