I just moved from new york, and have some ideas for the out of towner.
Union Square: To Eat: Republic, its on union square west. It's loud and they have communal tables, but the food is good and cheap. The drinks are a little expensive. For two without drinks your looking at 30, with drinks the skys the limit.
Show: Fuerzabruta (Brute Force) is really great for the under 30 crowd. You need to stand for about an hour, and it has the atmosphere of a rave with a trapeze show added to it. Normal cost is like 70 bucks but if you go looking in some bars you can find half price coupons.
Shopping: The Strand book store is amazing.
East Village: To Eat: Little Frankie's Pizzeria, its on 1st ave. It can be loud also but the food is really good. Expect to wait on the weekends. Bars: They are everywhere, DBA is great as is Karma.
Show: Ave Q! Has to be seen! This show is the mid 45s and 8th Ave.
Bar: Rudy' on 9th ave and 45th street. Classic New York City Dive Bar. The only place I will drink in Midtown.
I will write more of these, if anyone has any neighborhoods they are intrested in.