There is a cheap bus that goes from Bavaro/Punta Cana to La Romana town. Ask in Punta Cana and take a taxi to the bus stop. Buses leave several times during the day, you need to check the hours. From there you can take a taxi to the Dominicus Hotel.
Ditto, you can take a taxi from Dominicus Hotel to La Romana town and from there you take the express bus to Santo Domingo. It puts you into a terminal that is a dump, but in an interesting area -- Avenida Duarte -- a commercial street where everything is sold cheap. Just watch your belongings, you will be safe. Everyone uses that terminal that belongs to a union that does not allow competition.
Once you get into Santo Domingo, take a taxi (cost should be under US$3) to the Metro Bus station. There you can catch the bus to Puerto Plata town. If you are headed to Sosua, then get the taxi to take you to Caribe Tours bus terminal that goes to Sosua.
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If you are a real traveler and adventurer, then rent a car. To go by public transportation, lugging around suitcases, can be stressful. Pick up your car at the Punta Cana airport. See
http://www.dr1.com/national/
From Punta Cana to La Romana/Bayahibe is not a difficult drive. You just need to be sure to get off the main highway, where the detour to the Bayahibe area is. But there is a big sign.
My recommendation is always to have someone point out on a map where you are going, and every once in a while stop to ask someone. You will find lots of people who speak English, or can point you in the right direction.
Remember, ask when you think you may be going in the wrong direction and ask when you think you are headed in the right direction. Ask as you reach each point on the map.
Another pointer is to start off early so you get to your night time destination before the sun is down. Since this is summer, you have until 7 pm. Do not drive at night, as signage is not good, so it becomes less easy to ask and ask, as you can during the daytime.
At the Jumbo Supermarket in La Romana, purchase a good map and you are set to see the rest of the country.
Heading from La Romana to Santo Domingo, we recommend you stay on the main road, do not detour to the express highway. If you decide to take the chance, you need to go south at a cross point prior to crossing a bridge. If you do not go south at that point, you will find yourself heading to the north part of Santo Domingo, which means you trip into the city will take you an extra half hour or hour because of all the traffic.
Once in Santo Domingo, take the express lanes into the city. This will put you on 27 de Febrero Avenue. You need then to take Maximo Gomez, Abraham Lincoln or Winston Churchill Avenue crossings to go north and get on the John F Kennedy avenue that leads into Duarte Highway that will take you north to Puerto Plata.
The road north is easy, with the exception of the detour you need to take in Santiago to Navarrette and then to Puerto Plata. Again the signage is not good so you need to watch carefully for the road that goes to Puerto Plata.