By bus, boat, no train yet
Others will be more helpful about long-distance buses, but I can say that you can get anywhere you want to, at least by shared mini-van or taxi, at least during the day. These gua-guas are often slower, more crowded and hot, very cheap, very frequent and seem to go simply everywhere.
I did not cover your exact route in my recent two-week trip, but there is very likely an express air-conditioned bus route all or part of the way from PC to Santo Domingo. (what is that, like 150km?)
I would be astounded if Metro or Caribe bus lines could not take you the rest of the way to PP (~200 km?). Others can confirm the companies (and where the stations are, but you should be able to do PC to SD in three or so hours, and SD-PP in another three or so. If no-one here puts you straight, try the the lonelyplanet.com thorntree board for additional DR regulars.
By car? I didn't dare rent a car-- the local road rules take time to get to know, and Lonely Planet advises that foreigners are at great risk of being found guilty if any collision occurs, whether or not they are at fault. Regular posters can comment on that.
The risk of collision is definitely real-- two-lane roads west of Punta Cana (until you get half-way to Santo Domingo) are very busy and can be rough in parts. Lots of motorcycles and risky passing. Wide toll roads much of the rest of the way, though others can confrim what the roads are like on the last part of the route, Santiago to PP.
If speed is not the essence, and I wanted to see the countryside, and get to know the locals, I would go the less direct way via gua-guas and the ferry between Sabana de la Mar and Samana ("el barco/el bote" takes about an hour, not far from currently whale-infested waters -- a great side-trip until mid-March when the humpbacks head out)
You're talking a couple days there, with maybe 4-6 hours of bus/gua-guas a day. Three days would be more leisurely.
After just a two week trip, I now trust that a gua-gua goes where you want to go, and that it is waiting very close to you, ready to fill up and leave. The only problem is finding it -- the stations and routes are not always clearly marked, and if your Spanish is not great, there will be some confusion and frustration until you realize how extensive the system is. Not available later in the evening, from my experience.
It will be a few years, if ever, for the new train route from Santo Dgo. to Santiago to be built.
Does that help?
:glasses: