We did an excursion there a few years ago that took us to a tobacco factory, sugar cane plantation, a traditional Dominican home, and horseback riding on the beach. That's the only thing we did off property.
I will second the zip lining and jungle tour, FI and I went for his graduation a couple years ago and really enjoyed it. We also spent a lot of time on the golf course so that was the only excursion we did but it was fun to see everything on the drive up there and it's in a beautiful setting.
Re: Punta Cana