As hreis indicates, it is possible to take a bus most of the way, but this bus only goes once a day a 7 am from Puerto Plata and 7:30 from Sosua. After clearing customs, take a taxi to Sosua and stay over night. In the morning, have a taxi take you to the Caribe Tour terminal, arriving at the terminal before 7:30 am. You would stay on this bus to Sanchez, the last stop before Samana, then take a guagua from there to Las Terrenas/El Portillo. This guagua may be a pickup truck.
A more comfortable option would be to stay over night in Sosua, then take a taxi to El Portillo the next morning. More expensive than the bus/guagua, but much simpler for someone making their first trip to the area. Flying to El Portillo in the morning would be even faster.
I'd start by contacting your hotel and find out what they have in mind, then make your decision.