As everyone has suggested, take the coast road from Cabarete through Rio San Juan, Cabrera and then to Nagua. You need to pay attention in Nagua not to miss the left turn signposted for Santo Domingo and Sanchez. The sign post is right on the junction and easily missed, just past the hospital on the left. Cabarete - Nagua = 2 hrs.
Then take the toll road towards Santo Domingo. First toll is 213 pesos, second toll is 181 pesos, third is 53 pesos. Nagua - Las Americas = 1 hr. Watch your speed, there are AMET cops with radar guns trying to fine people for speeding. The speed limit is 80 kph. I was flagged down once. I was doing at least 120 kph. I didn't stop and kept driving thinking they might get me at the next toll booth. Guess what? They didn't!
At the end of the toll road turn left then immediately keep in the right lane heading down the underpass. At the underpass turn right, but take the left lane. This is will bring you out on the eastbound carriageway on the autopista Las Americas. Next toll is 30 pesos.
Follow the autopista Las Americas through Boca Chica, Juan Dolio, San Pedro and La Romana. At La Romana make sure you keep right and follow the signs for the new highway to Punta Cana. Tolls are 100 pesos, 50 pesos and 100 pesos. Las Americas - Punta Cana = 2.30 hrs.
Total driving time without stopping is 5.30 hrs. Total toll costs is 727 pesos.