Upvote:1
I have never done this in Dominican Republic, but I have in some other Central American and Caribbean countries.
It's usually fairly easy to rent a car and driver for a fixed trip to a specific place, or just for a fixed amount of time. It's best to book somewhat in advance, like a day or two. Assuming you are staying at a hotel or resort, try to book something through them. Almost every hotel knows of drivers prepared to do this. Make sure you agree a price in advance.
At a resort or hotel chain they will probably put you in touch with a 'private tour' company. This may be quite expensive (by the local standards) but very safe.
I've never tried to hire a taxi off the street. I would expect it to be possible, but more difficult, as those taxis are not expecting a trip that long. You could however approach a taxi and ask them to do something specific the next day. They would usually do that, and it would probably be cheaper than the hotel arranged tours. Again, be very clear about price up front.