I live in Northern NJ and can give you a few places...
1. KBID or KMVY - I prefer block island. My wife and i visit here a number of
times over the summer. The beach is fantastic, and I believe its the nicest around. If you go to Martha's Vineyard and want to spend time on the beach, I would recommend landing at Katama and taxing/parking all the way at the end. The beach is literally across the street. When you're done, throw your **** back in the plane and take a bus into town. $7 gives you bus access to the whole island all day long.
2. KLKP - Spend a day walking around town. Lots of nice little shops and great restaurants. I'd even spend the night if you can so that you can enjoy a nice dinner /few drinks
3. KASH - if you're looking for something fun to do, fly to Nashua and use the crew car to drive over to Sky Venture for some indoor "sky diving". My wife and I had a blast. Check Groupon before you go - many times they have a half off coupon. Make sure you call the FBO the day before to borrow the car.
4. KIAG - fly here and take a taxi over to the Canadian side of Niagara Falls.
5. KTGI - Tangier Island is like going back in time. Many people really enjoy going here. Some nice restaurants.
(if you do stick with just the hudson valley area - stop at Columbia County for gas... its under $4/gallon)