This certainly is not my field of expertise, but the Listin Diario has the most classified ads that concern Santo Domingo of any paper. There are also numerous real estate agents and most apartments and houses are listed with an agent. Agents work on a commission and generally have an interest in selling you on the highest price you will pay. It can be better to do your own negotiating in some cases or find a friend to do it for you. Also, make sure you have reciepts for any money paid. Even in honest relationships they can be very important.
There are many things to take into consideration. Location is important. This includes noise, guest parking availablility, crime rates and proximity to shopping, work etc. Perhaps someone who has lived here longer than I will chime in and list some of the best neighborhoods with their pros and cons. I know several of them, but not all of them. You might also check in the archives and I think you will find some imformation on that scattered about.
Electricity, water, maintenance of the apartment, and security are all important issues that will affect your price and your happiness. Many apartments have generators that serve the entire building. Some have generators, but only run them for certain hours of the day. Others might include inverters(battery power backup). Some may have problems with city water or flooding. It is always a great idea to ask some neighbors how things are. Some apartments have full-time guards for the entrance and/or parking area.
Since you are posting under "Anita," I am assuming you are a single female, please underline and highlight the importance of security. If you are broken into, or don't feel secure, you will probably end up moving to the apartment you should have rented in the first place. Moving is expensive and undesirable as is replacing stolen items.