Getting around Punta Cana is fairly easy if you decide to do what most people do – walking and taking taxis. However if you decide to be more adventurous, you could take local buses or rent a car. Your award will be that you will get closer to the locals than most people staying at the resort ever will.
If you go on excursions you usually do not have to worry about transport. Most excursions pick up tourists at their hotels and then drop them back. The transfer is typically included in the price of the excursion.