I lived there for one year. The places that I would try that are affordable are:
Casa Mono/Bar Jamon in Union Square - It is a Tapas placed that is frequently visited by top chefs
Tartine's - for brunch. South of Union Square by West Village. They have been filmed in movies and it is a quaint place.
Good Enough to Eat - a brunch place that is in the upper west side. They have a strawberry butter that is awesome. There is a line on the weekend's.
Corner Bistro - in the West Village is a great Hamburger place in the west Village. Been there forever.
Frank on 88 2nd Ave, New York, NY 10003 - is known for their lasagna. Very authentic. Does;t have a sign on the restaurant so it can be hard to find. Locals eat there.
Joe's Shanghai in Chinatown - is known for their soup dumplings.
If you have unlimited funds and time to make reservations - Per Se, 11 Madison Park, Daniel, Le Benardain.
Hope this helps