Hi berro,
Due to the condition of the roads and the extra space I think a jeep would be infinitely more comfortable and practical than a compact car.
There are a number of car hire companies at the airport and of course picking one up as soon as you land means saving on transfer/taxi costs.
You can google "cheap car hire in the Dominican Republic" and you will find them all listed there. I have used about 4 or 5 of the companies but have taken to using tropi-car -
Tropi-car Rent a Car. Puerto Plata and Cabarete Car rental with the best SUV rental fleet, located in Dominican Republic in Puerto Plata Airport - purely because they are good value, very easy to deal with and offer a personal service - the owner Naiyeni is very helpful. Make sure you are fully insured, won't you?
As for places to visit - buy one of the excellent travel guides available on amazon or ebay and have a read up - good for you for wanting to get out and about. The country and its people are wonderful and you will have a trip to remember I am sure. If you don't speak Spanish - remember to take a dictionary and phrase book - there won't be much English spoken once you get away from the tourism!
Have a great holiday
Rio