Nantucket is not the easiest (or cheapest) place to go. You can drive on the beaches of Cape Cod if you purchase an oversand permit. You will need to air down and carry some required equipment.
New Hampshire and Vermont have literally hundreds of miles of dirt roads that can be explored with stock Jeep. You might check out www.exploringnh.com
There are numerous clubs with members in the New Bedford area including:
South Shore Wheelers
USA Jeep Owners
Cape Cod Jeep Club
Ocean State Jeepsters
New England Jeepz
Clubs typically host 1 or more trail rides per month.