Things not to miss when you visit New York, Part 1: Great, Affordable Restaurant Food

Pizza

• John's Pizzeria (No Slices!) in Greenwich Village
• Lombardi's
• (Personal favorite) In favorable weather, walk across the Brooklyn Bridge (there is a quite safe, dedicated pedestrian walkway) and eat at Grimaldi's.

Jewish Delicatessens
Significant "endangered" factor here; changing food fashions are endangering the existence of delicatessens.

• Katz's at Houston and Ludlow St
• 2nd Ave Deli (now no longer on 2nd Avenue but on E 33rd Street between Third and Lexington Aves)

Bagels

• Murray's (warning: annoying big band music at website)
• H-and-H

Chinese

• Grand Sichuan, Ninth Avenue at 24th St