Where will you be in Puerto Plata? I can give you specific suggestions if I know that.
Speaking generally, one possibility is to take a Caribe Tour bus for the less than 20 mile trip to Sosua. But you would probably have to take a taxi to the Caribe Bus terminal and a taxi from the Sosua terminal into Sosua. The present cost for the latter is RD$150.
You could take a taxi to Sosua. As a visitor rather than resident, this is going to cost you about US$20 each way.
If you are adventerous, you can got to the central park and take the guagua to 'Sosua. These are vans and buses that load at the central park. Or if you are at a hotel on the highway between Puerto Plata and Sosua, you can flag them down as they pass. The guagual will cost you about RD$20. You can also take one of the publicos (public cars with red box on roof that says POP-Sosua). They, too, will cost you about RD$20.