My recommendation is not in the NYC area but if you do not mind heading over to New Jersey about 1.5 hour drive south... you can get to this amazing and peaceful camping at Belleplain State Forest. Plan a weekend with one of the 169 tents that you can rent out or just just pay per day for camp sites! You can go swimming on the north side of Lake Nummy (a former cranberry bog transformed into a 26-acre recreational lake), where lifeguards are on duty from Memorial Day Weekend (almost fully booked this year so hurry up) through Labor Day Weekend. Or take your pick from other recreational activities; boating, hiking, horseback riding, fishing and mountain biking are all options at a surprisingly low price.
If you prefer to stay up north with the cold, then I highly recommend going to High Point State Park. You can reserve one of the 50 campsites (no tent rent service) at this forested state park for a somewhat peaceful getaway (alot of bear activity so be careful) in the woods. I think you can still swim at the spring-fed Lake Marcia, or launch a boat at Sawmill Lake or Steenykill Lake. While you’re there, be sure to climb to the top of the 220-foot High Point Monument for sweeping vistas of the Pocono and Catskill mountains! Definitely instagram worth content.
P.S. I'd love to go to Bear Mountain but it have been fully booked since late January :'(
Hope that helps