The Museum of the City of New York is the place to go to find out more about the city. The exhibits have just about everything you ever wanted to know about New York City and then some.
They offer permanent galleries and exhibitions as well as visiting collections and shows, covering a total a tremendous quantity of perspectives of the city and it's inhabitants. There are often drop in creative workshops available and many other features to enrich the trip for smallish children.
Seasonal events and special programming are also regularly held here, with an events section purely for families, plus docents' choice guided tours help get the most education from your visit.