Many visitors think that they must take their children to very expensive Broadway shows in order to give them a great New York Experience. My now 8 year old great niece enjoys both worlds: Big Broadway shows and smaller theater performances geared to kids. I started taking R and her older brother C to shows at TADA theater when R was 4.
A couple of years later, I met R and grandma for dinner after they saw I forget what on Broadway. In the middle of dinner, R said "when are we going to TADA again". I said, "you've been to Broadway, why do you want to go back to TADA?". She said, "because we get to sit in the front row at TADA". Point is small theater experiences are both more immediate and less expensive for kids and teens.
Anyway, if you choose JR in the Fringe slicer genre field you will find a fun theater experiences geared to children and their parents. All take place in the 14th Street Y. There are many fun restaurants in the neighborhood.
If you put HIGH in the Fringe slicer genre field you will find thoughtful edgy shows at various Fringe Venues that will spark conversation among adults and older children of all ages.