I’ve spent four months in Santo Domingo and am there currently. The airport is about 30-45 minutes away from the Colonial Zone, and that would be where you’d want to go. It has the highest concentration of interesting places to see and things to do in the city for a visitor. The cab ride would cost you about USD$50 each way, so that’s something to consider. You could take an official taxi from the airport (one of the guys inside the airport with the “oficial” sign), and then call Apolo Taxi to take you back. Apolo is the go-to company for visitors around here.

Technically there are local buses and public cars, but I have never taken one without a local going with me. Daunting would definitely be the word unless you have lots of local knowledge.

However, if you take a cab and make it to the Colonial Zone, you’ll get a very nice chunk of history in one day. Santo Domingo is also a much different vibe than any of the beach towns, and it would definitely give you a more complete picture of the country.

Good luck with your trip!