Renting a car to drive in Santo Domingo is not a viable proposition for most people not used to this city with its horrendous, chaotic traffic. However, most rent-a-car companies offer chauffeur-driven vehicles that take the hassle out of the visitor.
Avis has a presence both at the airport and in the city. Unlike many local companies, their cars are properly maintained, and they provide roadside assistance, in case of an accident or break-down.
In any case, especially if you insist on self-drive, make sure that you take sufficient insurance to cover you in case of an accident, or unforeseen damage to the vehicle. Often, "full insurance" still leaves you with a hefty excess that you must pay to the rental company, and it may also exclude tyres, windscreen and other features of the car.
The comments above apply also to the urban areas around Punta Cana. The situation is slightly easier in the less crowded roads away from the main centres. Aggressive driving and the free-for-all attitude of many drivers, added to the large number of scooters that do not respect road regulations, makes for very defensive driving, and a high risk of an accident.
Tourists will do better to order a taxi (or a gua-gua for under a dollar) whenever they want to leave their hotel for shopping or going to another beach, while tour operators offer a better alternative to visit the local attractions. For instance, the entry fee for a car into Casa del Campo in Altos de Chevon is currently $25. Driving into Santo Domingo is a nightmare, while parking there is another!
On the positive side, roads around Punta Cana are generally well maintained and signposted. If you insist on hiring a car, don't drive at night: the crime rate is high and it is not safe!