The Met is a must—it’s one of the top collections in world, rivaled only by the Louvre and the Smithsonian collection. They’ve been redoing the place so things are in pretty good shape and there’s always an interesting exhibit going on.
Some of the smaller museums are also worth checking out, but first make sure you’re interested in their current exhibits or collections (and that the collections are not out on loan!). A few examples: the New-York Historical Society (the country’s oldest cultural institution!); the Museum of the City of New York; El Museo Del Barrio; and the Jewish Museum (it currently has an exhibit on Houdini). These museums are also generally cheaper.
NYmag.com and Time Out New York should be able to help you figure out what’s going on when.
You can visit Ellis Island which is a boat ride away and is a very interesting Museum on Immigration into the US