6 hrs? I don't think I'd go into town. Honestly, around the airports, there's not much, unless you're travelling during the Tennis US open. (well the queens zoo, ny hall of science, flushing park are all near LGA)

JFK...closest entertainment near there would be Belmont Park horse racing which is rarely open, and Rockaway park which is probably a pretty ugly beach.

Distance wise, Coney Island is not that far from JFK, but could take hour or more to get to from either airport.

As Blumie says, you could make it to the city, but I would recommend being on the train by 6:00 at the latest, and if its on a weekday, the trains are going to be pretty packed those hours.

Inside JFK terminal 1, there's a food court with nice views of one of the runways. Terminal 4 has some nice shops. You can also go around the airport and take a nap in one of the airtrains.