If you like museums, there's a somewhat off beat museum on the Lower East Side called the Tenement Museum, where you get a real sense for how miserable the conditions were that people used to live in there.
As an added bonus, the most amazing dumpling place in the world is located in a hole in the wall if you walk on directly south on the street leading out of the museum, and then at some point a block or two south on one of the smaller streets, hang a left and it's in that block. Sorry I can't be more specific, but it's worth a bit of a search.
Also for a bite to eat, I like Bianca's on Bleecker just before Bowery, one of those great Italian restaurants that only takes cash. If you're in the mood for yummy fondue, pop by Artisanal up around 31st street way, I think not far from Lexington. There's another good restaurant on Avenue B just south of 14th, can't remember the name though.
For more yummy and fast Chinese food away from the usual hipster haunts, check out the brighly-lit, unassuming place on the corner of 49th and 2nd.
There was a fantastic exhibit at the Grey Art Gallery near NYU called the Downtown Show, about the downtown arts and punk scene in the early 80s, but unfortunately I think it may have just finished. The link's here:
http://www.nyu.edu/greyart/exhibits/downtown/dthome.htm Anyhoo, if you can't go there, read "The Downtown Diaries" by Jim Carroll and "Please Kill Me," by I forgot whom, instead. Finally try the cheesecake at Yaffa Cafe on St. Mark's, and if you're looking for some places to stroll try the along the East River... just don't walk down Sutton Place, everybody's got an Akita.
