I would highly recommend the following places: The Museum of Natural History, Central Park, and the World Trade Center Memorial.
You can literally spend the entire day at the museum. There are extra exhibitions that you can pay for on top of the entry fee. Personally, I don't think they were worth the price. The only exception is the Hayden Planetarium. You should definitely fork over the extra admission for that.
When you visit Central Park, make sure to wear comfortable shoes. It's about a half mile wide and three miles long. There's a zoo, lakes, boat rides, a restaurant, or even just places to lay down in the grass. There are many statues and sculptures, as well as a castle and a 3,000 year old obelisk brought from Egypt.
The World Trade Center Memorial is a solemn experience. When I went, there were several people that were teary-eyed. Children probably won't be able to relate to the experience although there is no problem with bringing them.