Go for it! Then you can see La Romana town, with its many art galleries (such as Tambourini, and El Artistico), and of course Altos de Chavon (email for clearance to enter Casa de Campo through their contact email at
http://www.dr1.com/directories/hotels/casa_de_campo.shtml). And get to the Oasis Canoa from the airport, and then on to PUJ, where it is probable you could drop the car off. So you avoid many US$30s for taxi rides.
Check with National Car Rental that has offices in La Romana and Punta Cana. See
http://www.dr1.com/national
As for driving, just stick to daytime driving as there is little signage and less lighting – the opposite from driving in Europe. But, someone told me and I have to agree, that the reason there is little signage is that people have relied on asking, and getting an answer. So, do like Dominicans and ask when you think you are wrong and ask when you think you are headed in the right direction.
At the LRM airport they will give you a map to get to your resort. It is a basic go East and then turn South at the big sign that says Bayahibe. Then you take that road all the way to where it ends – avoiding a turn to the right that would take you to Bayahibe town – and you will get to your hotel.
Since you will have a car, visit Bayahibe beach town. When there, try to take an excursion to visit La Catalinita and Saona islands. Bring your snorkeling equipment. Also, take time to visit the public beach, right east of Dominicus Palace. I think that is the best part of the entire Dominicus/Bayahibe beach. If you are a diver, Bayahibe is one of the best places in the country for dive excursions.
To get to Altos de Chavon from your hotel you need to backtrack up the road to Bayahibe and take a left. That highway will bring you back to La Romana Airport, that you want to pass and continue until the road turns south and you see the Caribbean Sea. Shortly you will come to the main entrance of Casa de Campo. You will need to use this main entrance to visit Altos de Chavon. You will need to have called ahead to the Casa de Campo tel in the link above to get cleared for entering. Once inside, you could also visit the Marina de Casa de Campo. Shopping at Jenny Polanco/Bibi Leon (clothing and crafts) and Fiori store (for artsy handbags). Lots of parking.
From the Bayahibe crossroad, it is about a two hour drive to PUJ. You need to go East (take a right) once you get back to the big Bayahibe sign where you turned South to go to Bayahibe. Head East until you get to Higuey town. You will previously come to a cross in the road, take the left turn, not the right that will lead you to Boca de Yuma.
Continue East until you get to Higuey town. Then you need to pass the Basilica La Altagracia, the largest cathedral in the Caribbean. If you had time, you could make a stop and browse through it and pay homage to the Virgen de la Altagracia, the protector of the Dominican people.
Next you will come to a turn in the road and you need to head south, not north. This will bring you to cross Higuey and onto a road that you just need to follow until it brings you to the airport. Almost in front of the airport is Punta Cana Village shopping galleria. You can have a good lunch there. Variety of restaurants and shops.
Enjoy! Having a car truly opens up a new world of sightseeing.