Rental cars are expensive in the Dominican Republic. Unless you really must have a car of your own, there are good options. For distance travel from Puerto Plata, you can choose between Metro and Caribe Tours buses. Both companies provide excellent service at a very modest price.
For sort trips, you can take guaguas and public cars.
Taxis are also a good option; let someone else do the driving. You can negotiate your trip with the driver. Don't just pay the first price requested.
I've lived in the DR for more than 15 years without a car.