I would not recommend road cycling by yourself. The roads are not great but that should be the least of your worries.
Out on your own you will be an all-too-easy target for someone to knock you over and steal your bike. I am telling you this as a fellow road cyclist myself.
If you want to cycle in the DR, you will need to get in touch with guys who ride and get together with them. I have laid the groundwork for this myself, but in the capital, not out in Punta Cana.
If you bring a set of strong 32 spoke clincher wheels and some decent training tires(23mm, don't get cute with 19s, not on those roads) you should be set. A lightweight set of wheels will get abused quickly, and definitely forget about carbon rims and tubulars.
By the way, how is your spanish? If its' not great, contact the good people at
Iguana Mama - Mountain Biking, Hiking & Ecological Adventures They do some great organized off-road rides and they are very well-respected.
Whatever you wind up doing, let us know how it went and best of luck.