There are lots of cool things. Of the commonly known, The Museum of Natural History and the Planetarium are great. There may be some tours available also that go underground. Back in the early 70's I was a member of Local 147, the Sandhogs union. Those are the tunnel minors that "dug", more generally blasted the subways, etc. I only worked on one project-- a water tunnel that is as much as 800 feet down. We worked on the 800 foot shaft on what is now known as Roosevelt Island; it was Welfare Island back then. Once down to the bottom, we started driving a 30 foot diameter tunnel toward Manhattan under the East River and uptown to meet a similar shaft that was dug in Van Cortland Park in the Bronx. Believe it or not, my ears are still ringing. Imagine being in an enclosed 30 foot tunnel with 12 huge drills pounding blasting wholes into the rock! Here's a link to the underground layers (http://www.nationalgeographic.com/features/97/nyunderground/docs/nymain.html
). Notice the tunnel at the very bottom? That was us.
For all the criticism that New York takes, it is an amazing place. Did you ever think about how they managed to supply water to a population of that size and density? That's what that tunnel was for. One truly remarkable thing is that they have always stayed ahead of the need. Anyway, I do not know what underground stuff is open to the public, but it would be fun. Here's something of interest, http://hubpages.com/hub/Subterranean-New-York-Fort-Totten-Tunnels.
Notice the reference of Captain Robert E. Lee.