Any of the state parks in NJ or NY are great choices. You can look on the NJ State Park website and they have a list of free activites going on as well. For example, Jenny Jump has an observatory where you can go look at the stars on Saturday night, but there are other activities as well. Plus kids generally just like to roam free, walking or riding their bikes around a safe place.
My kids are picky eaters, so often we camp by a place where we can run out to grab something to eat. But snack food around the campfire is what makes the trip, think smores, marshmellows and Jiffy pop (the kids will be amazed that the popcorn didn't come from the microwave). In the tent, bring a glow stick and crack it open as a night light, provides enough light for them to feel comfortable sleeping, but not enough to keep you awake. Have fun and don't stress.