We stayed at the Westin on Times Square (actually just off it). Characterless but perfect - clean, efficient, great location, not too pricey.
Took a half day walking tour with Real New York Tours - brilliant and fun, stuff you'd never get from a guide book. Gave us lots of tips for the rest of our stay, too.
Staten Island Ferry - free and a great way to see the Statue of Liberty close up without queuing.
Top of the Rock observation deck for great views and photo opportunities. Can take photos of the Empire State building rather than being up it and not able to see it, if that makes any sense!
On a rainy day, the Madame Tussauds was good fun - you can have your photos taken with the Obamas in the White House which is a great family photo!
MOMA and Guggenheim if you are at all interested in art. Even if you're not, actually!
Central Park, obvs. Grand Central Station fascinating and beautiful, with a great oyster restaurant (serves lots of other stuff too).
Lots to see just walking and looking. Our kids were about 9 and 11 at the time I think but I've never met a kid of any age who didn't love it. Teens even more so as it feels like you are in a film set the whole time.
Loved it. Am taking my dear mum there this year so need to start planning for that………..