We recommend that you drive, only if you have four hours of day light. We do not recommend that you drive at night. Better to pay a taxi.
If you go by daylight...
Head East all the time. The key is to ask, ask and ask. People will be helpful. Just keep on the main roads ask for each town.
The towns you will pass are:
Boca Chica (the highway actually passes north of this town)
Juan Dolio (the highway also goes north of this town)
San Pedro de Macoris. Gets a bit tricky here because this gets inside the town. Here you want to go in direction of the Zona Franca. Once you get to the Zona Franca, you turn East again (left) and continue on the main highway.
La Romana. Here it is a bit tricky. Here you want to go in direction of Casa de Campo. Once you pass the entrance of Casa de Campo you continue on that same main road to Higuey. May I suggest, if it is still early, that before continuing on to Bavaro, you make a detour to see Altos de Chavon. After passing Casa de Campo, it's the next big turn to the right. They don't have a big sign, so when you see a road turning to the right, ask to make sure.
La Romana Airport. El Limon shopping center. Dominicus entrance. You need to pass all these continuing East all the time.
Higuey is again tricky because it is a town. You want to go in the direction of the Basilica de la Altagracia. Once you pass the Basilica when the road ends you turn south, or to the right.
La Otra Banda. A very picturesque town on the way to Punta Cana-Bavaro.
Now you want the road to Bavaro. Just ask for the Allegro Bavaro Resort. There will be road signs all over the place advertising the different hotels.
This can be a fun trip. We recommend that you not be timid, and ask, ask and ask to make sure you are going in the right direction.
The entire trip, without stop, will take about 3:30-4 hours.
Also see another contributor's description of part of the road, but the other way around, from Punta Cana-Bavaro to Bayahibe/Dominicus.
http://dr1.com/forums/showthread.php?threadid=15482