I would guess there are places in La Romana where they won't be able to provide you service. I am also sure they won't be able to give you a map that shows where : ) But, if you give them an address, they can tell you if service is available. They have a toll free (sin cargo) telephone number you can call for info. It is on their web page. If your Spanish is not good, just say no hablo and they will transfer you to a guy who speaks English. They are very helpful.
But, of course, the real customer service is just like everything else here - they don't show up on the day they say they will to connect your service, the guy forgot his tools on the first visit, but came back the same day with tools. Etc.
I do know Claro also has service in La Romana, but looking at Tricom television and internet together Tricom was a little better for us. TV channels, price etc. We pay $2450 pesos per month for "superior" tv (no HD) and 4 mb internet. There is one higher tier of TV with HD, more channels, etc. Costs more. They also have a kiosk at Jumbo, you can ask questions, or order service. The people on the phone know more though.
I don't know much about other providers, I do know Aster does provide tv service, but apparently not internet. I think their tv would be ok, they have a web site. Appears to say internet only is Santo Domingo, but I have not talked to them.