Hi Charlotte,
We live about 3.5 hrs from NYC and try to visit often. I recommend taking the subway to Brooklyn, then walking back. Also on the Brooklyn side of the bridge are great views from the Promaenade and the Transit Museum, which is an old subway station. Very interactive, you can get in the old trains, etc.
http://mta.info/mta/museum/index.html
My favorite museum in NYC is the Tenement Museum, which has different tours by time period and ethnic groups. Your kids are old enough to appreciate either the children or adult tours. The tour groups are small and should be booked in advance. Also in this neighborhood (Lower East Side) is the famous Katz Deli, Donut Plant and Economy Candy.
http://www.tenement.org/
If the weather is good the Highline is a fun attraction. It is a park made on an elevated rail line (similar to the Promenade Plantee in Paris.) You can begin or end this walk at the Chelsea Market which is a fun place to pick up some snacks or lunch.
Other hints:
The TKST booth at the Southside Seaport has way shorter lines than the one in Times Square and is near the Intrepid, not far from the Brooklyn Bridge. You can also get discounted tickets online via Broadway Box.
If you plan to do the Statue of Liberty and Ellis Island, definitely prebook these tickets (the Crown tix sell out fast.) The museum at Ellis Island is amazing.
Using googlemaps to plot the sites you want to visit is a good way to plan days. You can see what things are near each other. Then use the "Search Nearby" function to find restaurants.
John's Pizza is great and the location near Times Square is an old church. It would be a good place to get dinner before your show.
We are headed back at the end of March and can't wait. Have a great trip!