I live in NYC and agree with Pisticci, Red Rooster, Dinosaur BBQ, Veg Dim Sum, Gramercy Tavern ($$$$, years ago way before MMM was in my life- these days I'd prob skip it), Porchetta, Luke's Lobster, and Celeste. I would add for excellent relatively cheap eats:
-Setagaya (ramen house) 34 st marks pl (2/3) order the shio ramen- quick lunch or dinner
-Cafe Asean (Thai) W. Village-- nice ambiance for dinner and great, relatively inexpensive entrees and cocktails
-The more casual David Chang place in the East Village for pork buns-- will think of the name...
-Russ & Daughters (famous bagels and fish)-- order the Super Heeb for an
unforgettable Lower East Side breakfast
-J.G. Melons (burgers) Upper East Side- if you're on Museum Mile and want a delicious affordable dinner, head to this gem on 3rd Ave.
-Maison Kayser (French bakery and lunch) UES- ditto but for bfast or lunch.
I could give you neighborhood specific recommendations if you know where you're staying, too.