Or maybe "Ye Olde New York" is more your speed? Check out a bar from the days of prohibition: Chumley's on 86 Bedford St ... look carefully, since I'm not sure that there are numbers on the front door! The food isn't that great, but if you are looking for a burger and a draft, this place is a NY classic.
Burgers not your bag? Pearl Oyster Bar (18 Cornelia St) is a great find for Maine lobster rolls, a big bucket of steamers and home-made blueberry pie. The chef at Pearl grew up in coastal Maine, and her menu changes daily depending on what the fishmongers pull in from the ocean. They don't take reservations, but since they have expanded their space, the wait is usually just 10-15 minutes, if there is one at all.
Employees Only (read about it here. (510 Hudson Street) is very pretty (the interior and the crowd). Not your typical West Village haunt.
NoLIta (North of Little Italy)
This is my favorite neighborhood in the whole city. You won't find a Starbucks, Gap or Banana Republic in this tiny little slip of Manhattan.
Cuban Food! Cafe Habana (17 Prince St) is a GREAT place to grab a Cuban-style breakfast or lunch -- the cafe con leche and corn on the cob is not to be skipped! Huevos rancheros are delish, and a steaming plate of black beans and rice will fortify you for a chilly day of shopping in this gem of a neighborhood.
Sake! Once the sun sets and you need a place to settle in while you figure out where you're going to eat dinner, step into Chibi's (28 Prince Street? Not sure the exact address on this one, but it is connected to the Kitchen Club) for a lovely glass of sake from their extensive sake menu. If you are lucky, you might even get a chance to meet Chibi! (the very friendly and pudgy French Bulldog the owner Marja named the place after) I love how low key this place is -- you can almost always find a seat, and they will never shoo you away if you like to linger.
Continuing with the Asian theme ... if you are looking for a light bite, try Rice (227 Mott St). They don't really serve full meals here -- just lots of little plates, and most are rice-based. This sliver of a restaurant is usually busy, and for several reasons! Great location, inexpensive, hip crowd, lots of menu options.
And then catch an independent movie just up and across Houston at The Angelica Theatre.
Random
Paul Smith has a plucky store on 16th and 5th. Agnes B is on 16th btween 5th and 6th if you have a shopping bug.
Ling's is the place to get a facial (on 16th), Lush is the place to get a pedicure (Thompson St.)
City Bakery (18th? between 5th/6th) has hands-down the best thick hot chocolate with home-made (!) marshmellows!