First thing you need to do is walk out to the highway. If you have a lot of luggage, this can be a problem. It isn't a long walk if unencumbered, but if loaded down with luggage, it will be a long, hot walk.
Since you want to go to Rio San Juan, cross the highway and wait. Vehicles passing on the airport side of the highway are going to Puerto Plata.
You have two options for public transportation. One is the guagua, which often is a van but may be a bus. If you are lucky and get a bus, you will be more comfortable than in the van. The other option is a publico (public car). The cars going from Puerto Plata to Sosua have a red box on top. They will take you to the cars that have a blue box on top. Change to one of them for the ride to Rio San Juan. The normal load in a car is two paxs in front seat with driver and 4 in back seat. However, you can pay a double fare and have more room for yourself.
The cars will get you there quicker because they have fewer paxs and less room, so don't stop nearly as often to load and unload paxs.