You have time to make a Caribe Tours early bus to Santo Domingo and from there take another bus to Samana. Figure 1:30 minutes to Santo Domingo, and 4 hours to Samana. I suggest you take the time to visit the Centro Leon in Santiago, then take the bus to Santo Domingo, overnight in the Colonial City and see the sights, and then the next day take the bus to Samana. From there you can take a minibus to Las Galeras.
There should be other minibus service (less comfortable and probably making stops) from Santiago to Samana, (about 4 hours), but not sure who offers this. Probably someone else could suggest.
Other option is just to rent a car at the airport. Head south to San Francisco de Macoris and then east to Samana through Nagua. Once you get to Sanchez, you know you are there. Then comes Samana and where the road ends is Las Galeras.
There is no scheduled domestic plane service. You would have to charter a small plane, and unless there are several of you, that could be expensive. If money is not a problem, the answer to your question is: the plane. Will get you to Arroyo Barril airport, off Sanchez. From there you take a minibus to Samana, and then one to Las Galeras.
Air Century, S. A. (DR1)
Checked on the Linea Santiaguera (809 576-0790) mentioned in the other thread. Was told that to Las Terrenas they have buses that leave Santiago at 6:30 am and 3 pm. And to Samana city they have buses that leave at 8, 9:30, 10:30 am and 1:15, 2:30 and 4 pm. Was told that the trip takes about 5 hours because of all the stops made on the way.