TNJ1996,
When you get to your resort, they will have a number of day trips that you can do. All are great.
What I would recommend for Santo Domingo, is to take the express bus at 7:30 am at the Bavaro Plaza. When you get to Santo Domingo, rent a taxi for the day, and have the taxi person take you to the sites to see in the Colonial District. This place is worth seeing. I was able to find someone who wanted 300.00 peos, however, he was such a sweetheart, I did pay him more for his time, and treated him to come along and see the sites. Some of them know a little english. There's a taxi guy name Mateo or his cousin. His cousin is the person that I got. Just mention the name Mateo and anyone can tell you where he's parked.
Renting a car is great, to go to the towns of La Romana, Higuey, Boca Du Yuma. To see the culture of the island. Rent you car from where you are from, it's cheaper than reserving in Punta Cana.