Tips from our recent family trip to NYC. My kids are about the same age.
We saw School of Rock, good show for kids/ family. If you or your family is not overly "in to Broadway" this would be a good option.
Get a city pass before you go. You can do a little research and pick the one that allows you to see what is on your list and save a little. Top of the rock has better views then Empire. We did both. Do top of rock late at night. There are three levels and we were able to go to the very top with out paying extra. Again, awesome experience at night.
Going to the Statue of liberty will kill 2/3 of a day, not really worth it in my opinion, YMMV.
Central park is worth the time.
My kids loved Dylan's Candy Bar, FAO Piano from Big, NBC studio tour, checking our trump tower, and Times square at night (hit up after a Broadway show)
Natural History Museum is good, not great. The Met is awesome. We did not do the Aircraft carrier museum (forgot name) and regret it not doing it. Its a little out of the way.
Game at Yankee Stadium is a must do
We stayed at The Quin, one block south of Central Park near the Plaza and Trump hotels. Great location, and a large room for NYC standards. Would recommend if not using points for hotel.
Food is going to cost you a fortune. I still get pissed thinking about what I spent. Prepare for $125-$175 lunches with no alcohol. Dinner can hit $200 fast.
Joe's Shanghai for really good Chinese and is affordable.
Found The Smith on our last morning. Great food, open Breakfast lunch and Dinner.
Balthazar is fantastic and worth every penny.
Was not impressed with ABC Kitchen-
Start booking reservations ASAP for lunch and dinner.
Download and set up Curb app. Lets you pay for cab through app. Cabs are generally easier then UBER/Lyft unless you are heading across town.