More detail
I thought I'd add more.
I agree with everyone that you should drive and, most of all, enjoy your trip. However, there are things to consider.
ALWAYS lock your doors while you are in your car.
Do NOT open your doors for anyone and do NOT give a lift (bola) to anyone, including women on their own. Even if Police or Army try to stop you, it is best to be very cautious. If there are lots of people about in a town/city, then the Police/Army probably are who they seem to be. Where there are no onlookers, assume they may not be legitimate and try to deal with them only through a partly open window and only if you have to. I can't think of any reason why they should require you., legitimately, to get out of the car unless you've been arrested and a police car is called and arrives. (While waiting, call someone and tell them what's happening.)
You're a woman (yes I know that's obvious!) so you probably drive less aggressively than most men. However, do NOT be tempted to be aggressive. Always give way - you lose nothing but a few seconds of time but could easily save days or even your life. Dominicanos do not assume they need to give way ever - not even at red lights or when overtaking on blind bends.
NEVER drive so fast that you cannot stop no matter what happens. if you hit someone who steps out in front of you, the Police can put you in prison while investigating the accident or can ask for a consideration (if you see what I mean!) so as to avoid prison. Minor offences that Dominicanos are allowed to get away with (e.g. illegal parking, driving the wrong way down a one way street, running a red light, turning illegally, etc.) will attract immediate attention and again usually result in you needing to give a consideration (!!!) unless you want to spend hours waiting for a police car to take you to a police station to explain yourself. So drive so as to avoid the possibility of even accidentally offending, such as passing a red light. (I've witnessed teh Police stopping foreigners for passing a green light that turned red just afterwards!)
Carry a working mobile phone and ensure it is programmed with or that you have with you the number to contact your embassy (ideally 24 hour emergency contact). make sure you have the number of the car hire firm to hand.
Accept any optional insurance for accidental damage and theft. (On my first visit, the angry and drunk husband of a lady in a house \i was visiting rammed my car to show his anger at his wife - who I didn't know and wasn't visiting. Dominicanos are not logical. On my second visit, a bus removed the left hand side of my parked car because he was driving too fast and lost control. Dominicanos have little or no driving skill and are almost always aggressive drivers.)
Do NOT assume out of town driving is safer than in-town. In-town driving is obviously dangerous and there's tons of traffic to remind you so. The problem is that out of town there is little or sometimes no traffic to remind you how dangerous it can be. Dominicanos will do ANYTHING on a road. Expect them to drive the wrong way on your side, overtake when you are making a turn, park alongside each other (no limit to how many cars will do this), indicate the opposite to what they intend to do, not indicate at all, overtake on blind bends (so always be prepared to stop), etc.
Remember, roads have holes (sometimes there are more holes than road), the side may have a concealed ditch or drain, country roads in the mountains can subside if you park very near the edge (one of my friend's friends died like this), sleeping policemen (concrete humps) are rarely well signed but can cripple your car if encountered at speed - they are usually near plice stations, army posts and schools but may also be elsewhere.
Dominicanos do not understand the concept of risk. if there is a gap, they will attempt to drive through it, at speed, irrespective of what you or anyone else might do. This includes overtaking on the emergency boundaries (both sides) of a motorway. If you are signalling or even starting to overtake on any road, it will not stop most Dominicanos from overtaking you so as to get in front.
Cars that overtake you may immediately stop or turn in front of you.
It IS legal to turn right on a red light unless there is some sign saying not (I've never seen one).
If any mountain locations are on your list of places to visit, the roads can be covered in thick cloud from very early in the afternoon. Again in mountains, be cautious, drive well clear of the side with a precipice as there can be unexpected holes or the road can give way. Still in mountains, avoid driving is heavy rain as the mountainsides can subside or there can be rockfalls. If beginning a trip in the mountains, fill your tank first, fuel may be more difficult to locate. Buy the best fuel available because all fuel here is dirtier than you're used to so can cause problems.
As others have said - simply do NOT drive at night unless you are very experienced and have relatively advanced driving skills or unless you are driving a road you know VERY well (but STILL be very cautious - there are many additional problems at night that nobody has described).
Tolls on the motorways are 30 pesos. before you drive a motorway - go to a bank or buy something in a shop so that you have 30 pesos in change. Then you can just throw it in the American-style bucket container by the gate - no need to get change or stop at the preceding kiosk. Alternatively, the kiosks change money (try to have a small bill, say 100 pesos, never present a $2000 pesos bill as they will not change them) which you then throw in the bucket - they give you the 30 pesos separate from whatever else they need to give back). Avoid the lane that says Paso Rapido as that usually allows passage only for people who've paid in advance - once at that gate, the only way out without a pass is to reverse! It's unlikely that you'll do any motorway journey in a single day that requires more than one toll but ideally allow for two tolls in case you can't get change (moneda) anywhere.
Never park on a corner as it's an invitation for someone to crash into you.
Motorbikes and scooters (pasolas) are 1000% worse than cars. Assume they will always attempt to drive in a straight line (so they almost always cut corners) except when on a straight road, when they will generally wander down the middle weaving from side to side. Motorbikes and scooters also have a preference to never stop. At a corner they simply turn, without looking or reducing speed, hoping anyone else will take avoiding action if necessary.
If you do get involved in an incident - your car rental company will want the Cedula (Identity Card) number of those involved in addition to any other details. A photocopy is even better but is probably too much for a visitor to be able to get unless already at a Police Station (though usually the Police will not have a photocopier - there may be one nearby).
If there are 3 lanes (motorways) the best land is lane 2 as fewer people park in or queue to turn left or right from that lane (and yes, I did say "fewer", not "nobody").
If you are passing another car or lorry, always hoot. Even so, assume they may be listening to deafening music so may not know you are passing (mirrors, if they have them, are mostly decorative as far as Dominicano drivers are concerned). Hoot if passing any other vehicle that has already demonstrated a tendency to weave or overtake unexpectedly or if you see that they may need to soon - because they will unless you give them significant cause not to.
To avoid the annoying window washers at traffic lights, many of whom make your windscreen dirtier than it was, just switch the wipers on as they approach. To ignore beggars, sellers and the like, look ahead and do not look them in the eyes. If they clean your windows without your consent you do NOT have to pay. If you ever feel the need to pay, 5 or 10 pesos is sufficient, never more.
If asking directions, try and ask more than one person in separate locations - before you've tried to follow anyone's instructions for too far. People who do not know the answer to your question will often tell you the answer - passing their ignorance to you as if it were truth! Also, a popular game here is to misdirect - people may deliberately tell you the wrong directions. In any case, use a map when possible. Dominicano direction are based on use of "above", "below" and other equally vague terms such as "there", all accompanied by uninterpretable hand signals. It's all meaningless, even to other Dominicanos!)
Despite all of the above, most of the foreigners who live here do drive and survive so there's no reason why you shouldn't. Just remember, always be cautious, always be alert and attentive (WATCH THE ROAD) and always be prepared for the worst to happen.
Have a safe and great time.
Cheers.