Basically what arn3696 said. KFRG/Republic has plenty of rental options (Long Island Aviators, Ventura, Nassau Flyers, Academy of Aviation) and is not too far from NYC. On a good day, it'll take 45mins to drive from Manhattan to FRG. Options in NJ are similar (KMMU, KCDW).
Plenty of 172's available for rent.
It's easy to fly around NYC. You can fly up and down the Hudson River in the SFRA without entering the Class B, or if you want (and I recommend), flying inside the Class B and you can fly right across Central Park.
As far as other places to fly within 200NM:
- Montauk (the far end of Long Island), great place to go for lunch (mainly summer)
- Martha's Vineyard or Nantucket (mainly summer also)
- Block Island (yet again summer oriented)
- Boston
- Atlantic City
- The Berkshires, Catskills or Poconos
- Follow the Hudson River northbound for a bit and enjoy the scenery then turn around