Other than the capital, I don't go anywhere
And even then the last time was maybe 8 trips ago ( the whole Colonial Zone thing, which I recommend). I went shopping this March for a DVD player at the Plaza Lama but that was for a resident who is a luddite.
I am thinking a side trip to Santiago to experience the Metro Bus, maybe. And the stuff like Brugal or Jimines factories.
But basically I go breakfast, chores, pool, lunch, beach, dinner, chatting with friends, repeat. A different beach is still a beach to me, so travelling around the country side holds no interest.
I know others may have more defined things to see like caves and streams and so on, sorry , I don't. I would say, in the spirit of AI avoidance, skip the "jungle tours" and such with gigantic trucks and 2 gallons of rum each that are really aimed at the AI tourist. At outrageous cost.
I think Rob here in DR1 had a great Colonial tour, for a bit of a cost but to me sounded ideal if you really want to know the history and see the inner city of old.
If you are cool with speaking a bit of Spanish (or a lot), hire a car and driver, set the rate in advance and head out one direction or another for a few hours - go to La Romana, San Pedro, see where people live, visit a batey ( I visited the one in San Ysidro with an American who speaks creole and a Haitian driver).
Sometimes just seeing the different things is a lot of fun. Like a Dominican grocery store is interesting with products you just don't see. I had fun going to a hardware store in Coquito to get some nuts and bolts - had to learn the words for them first, the find them in the store. 5 pesos of fun.