Try going to Washington Square Park in the Village on a Saturday afternoon. You'll see a huge cross section of humanity there, old folks playing chess, street performers, skateboarders, comedians, maybe an anti-war protest or two, loads of students from NYU, sometimes an impromptu jam session.
As risque as it sounds, The Museum of Sex always has an interesting exhibit going including a permanent exhibit that is an homage to the seamier side of New York that no longer exists thanks to Rudy Giuliani, though oddly enough, he was a big part of this museum becoming a reality. You'll see ancient artwork there as well as pop art and artifacts and don't miss the gift shop.
If you want to hit the museums, NY City Pass is a great deal. For $63.00 you can get into 4 of New York's best as well as the Circle Line Cruise around Manhattan and the Empire State Building Observatory at your own pace over a 9 day period.
To get a bit of the feel of the literary history of New York, stop by the White Horse Tavern (not far from Washington Square Park), where Poet Dylan Thomas was a regular (there's a rumour that his ghost is still there).
When in doubt, once you get to New York pick up a copy of TimeOut NY magazine. It's a weekly that's always full of things to do that will appeal to every one.