Morristown is in Morris County - northwest NJ. I live in the town right next to it (Morris Plains).
Land of Make Believe is fun for the ages of your kids and only about 45 minutes from here. Easy ride - 80 west and about 2 turns off the exit. The water area is fantastic but can get really, really crowded on a weekend or Friday. The rides are fun too. Its a campy-type of place. You'll be safe in June crowd-wise. Our schools don't get out until the end of the month (we're the 19th) so that means the camp buses won't be there for a few weeks into July.
Point Pleasant is just about an hour away. Again -another easy ride. South on the Parkway until you see the sign! Jenkinson's is there with a nice aquarium, the beach, boardwalk, etc.
Let's see... there are lots of things to do around here - hiking, historical tours (amongst other landmarks, Washington's Headquarters is in Morristown), beautiful parks, some GREAT playgrounds. NYC is about 45 minutes away and of course, there is a ton to do there! You can even grab the train right in Morristown. Its really the best way to get into the city if you're not familiar with or comfortable driving. Museums, shows, tours - all so close in NYC. The aquarium at Coney Island is awesome.
Dutch Wonderland and Hershey Park are closer to 3 hours from here, Great Adventure is a good 2 hours. Rye Playland is icky (although I haven't been there since I was very young). Depending on when you're coming you may hit a street fair here or there with rides and fireworks.
Good luck! Have fun!