From Georgi Goldman...

Go to the TKTS booth in Times Square or down by South Street Seapport for discount tickets for Broadway shows - both matinee and evening performances.

http://www.tdf.org/TDF_ServicePage.aspx?id=56

If you are downtown, go the Bodies exhibit (though it would be a waste to be inside if it's a nice day outside)
http://www.bodiestheexhibition.com/

Get a burger at t he Shake Shack in Madison Square PArk - delicious.
http://shakeshacknyc.com/camera.html

It's always nice to go to Central Park - and there are a ton of things to do in the park
My favorite thing to do is get a row boat and row in the lake - the boathouse restaurant is nice too:
http://www.centralpark.com/pages/attractions/loeb-boathouse.html

And the Central Park Zoo - http://www.centralpark.com/pages/attractions/children%27s-zoo.html

ANd you can take a horse and buggy ride.

And if you enter or exit the park by Columbus Circle on the southwest corner, you can shop at the Time Warner Center.

I could offer a ton more suggestions if you let me know what sort of things you may be more interested in
Summer is the best time in NYC for free events and Im sure there are tons going on - Id pick up a Time Out New York and/or a Time Out New York Kids (or just browse one on a newsstand) for ideas. Also, sometimes freenyc.net has some cool things listed.

[ed. I found Time Out NY Kids online at tonykids.com. Good stuff. Thank you for the thorough recommendations, everyone.]

I recently went to Ellis Island and thought that was pretty interesting. South Street Seaport is fun. Chelsea Piers has some fun stuff (like year round ice skating). And you can't go wrong with the buggy ride through central park. Always love to try and catch a Broadway show if Zoe can stay up that late - you can just go to the TKTS booth in Time Square in the afternoon and see what's available

What's that deli with all the sandwiches that are as big as your head and named after famous people? Yeah, GO THERE!! GOOD FOOD!!

From David Neuman...

Empire State Building, Statue of Liberty, Central Park Zoo and hansom cab (horse and carriage) ride, Rumplemeyer's ice cream on Cental Park South, FAO Schwartz store in midtown.

From Steven Blumendfeld...

Natural history museum always a winner....carmine's for family italian 42 and broadway for dinner....

Statue of liberty - lots of walking
Moma museum - may not hold interest....
Lots of things to do from 'alice in wonderland' and boats in the park to broadway shows to the holographic musuem (14th if memory serves)

going up the Empire State Building if they have not. Taking the ferry to Staten Island and back, looking at the skyline and the Statue of Liberty while on board. Walking from Battery Park to Greenwich village along broadway, stoping to see the World Trade Center, Chinatown, Little Italy, Canal St and Soho.

If you go to the Empire State Building, make sure to eat on "Korea Way" (32nd st, west of 5th avenue).