If you are fairly adventurous, have rented abroad before, and are not timid, go ahead. It makes sense to rent a car if you have to pay the about US$100 taxi fare to get to Punta Cana. If you know some Spanish, even better. Don't do it, though, if your flight gets in the afternoon. It's at least a four hour drive (five hours accounting for some stopping and getting lost). You don't want to drive at night.
If your flight arrrives in the afteroon, consider overnighting in Santo Domingo and leaving the next day. We recommend staying at one of the Colonial City small hotels.
The car rental should provide you with a good road map and mark the main points.
The secret to getting anywhere in the DR is to ask, ask, ask to make sure you are in target with the points marked on the map. When you think you are wrong, ask. When you think you are right, ask. People will be helpful showing you the way. Worst comes to worst, you can pay a motoconcho to lead the way out of the difficult places, like crossing through San Pedro de Macor?s, or Higuey. That will only cost you about less than US$1.
There are few signposts, but it can be done, and will be rewarding because you will see so much more. You can drive through Boca Chica, then drive through Juan Dolio, and then drive up to Altos de Chavon in La Romana, and then to the Dominicus beaches, before heading out to Punta Cana. Of course, you can go direct and then use the car for day time excursions to these places. In the Punta Cana area, there's the Ponce de Le?n house, and there's an Indian museum.