There is a lot going on this summer at the sometimes forgotten Queens Museum nestled in Flushing Meadows-Corona Park and well worth the trip if you havn't been. This summer's events at the museum include an international film, dance and music series, free archery, scavenger hunts and lots of art making and viewing opportunities with the family.
The Queens Museum focuses on 20th-century works and is located right next to the Unisphere which is that awesome retro steel globe sculpture that was made for the 1964 World's Fair, check out Forgotten NY for some cool vintage photos and info on the Fair. And since school groups are the largest single community that visit the museum each year, they really appreciate and cater to kids and their staff is uniquely qualified to work with children.
As for getting there, while it is a bit of a trek, you can get there on the 7 train and the trade off is that once you are there its usually not too crowded and all their summer activities are free with admission ( $5 for adults, $2,50 for kids. Kids under 5 are free).