La Follia is a pretty new Italian place on 19th St and 3rd ave. It has a wine bar section and restaurant and is definitely cozy. Pastas and main courses range from $11-18 and they have a good selection of wines by the glass or 1/2 carafe.
226 3rd Ave, New York, NY 10003
Look into the East Village. Much easier to eat cheap(ish) there than in other nabes near union sq. Some that I've been to are Jimmy's No 43 (good beer selection and solid eats -- really enjoyed the pork belly) and Cacio e Vino (can work within your budget if you order wisely). If upscale pizza works for your purposes, then Motorino would be a good option.
Cacio e Vino
80 2nd Avenue, New York, NY 10003
349 E 12th St, New York, NY 10003