JG Melon has a great burger, it's on 74th and 3rd Ave. EJ's Lucheonete or 3rd and 73rd, but lines can be long on Sunday. Fig and Olive on Lexington bet 62/63 is a bit more expensive, but it is very good and still within your price range.
1291 3rd Ave, New York, NY 10021
Fig & Olive
808 Lexington Avenue, New York, NY 10021
1271 3rd Ave, New York, NY 10021
I live on 63rd and head to a few spots in the area. Try Cabana on 61st and 3rd. It has Cuban/Latin food with really big portions and reasonable prices. They don't take reservations though and the wait can get long. Finnegan's Wake is our favorite Irish Pub, good food and cheap (don't expect a happening crowd though). A La Turka is good Turkish on 74th and 2nd and Go Burger (BLT burger spot) is on 75th and 2nd which I like a lot as well.
1361 1st Ave, New York, NY 10021
A La Turka
1417 2nd Avenue, New York, NY 10021
1448 2nd Ave, New York, NY 10021