That can be an interesting drive!!
You will have to budget about 8 hours for it.
If you stick to major roads, drive defensively, and use your head, you should have no problem whatsoever.
Us "old hands" always like to warn "newbies" about the very real hazards of driving here, and I totally agree.
The route is very easy to Santiago. Then you have to choose between the Tourist Hwy or the Duarte Hwy. Tourist Hwy is shorter, lots of curves, plenty of views, less traffic.
Duarte Hwy is busy, under construction, heavily travelled.
To take Tourist Hwy, after the light at the entrance to Santiago (across from Plaza Monumental), look for a sign directing you to Puerto Plata--bear right off the hwy, but keep going parallel to the hwy, taking the second right, towards the north. At the second light on Ave Estrella Sahdala (Hach?), you have to choose:
Go straight to Duarte Hwy, or turn right for Luperon Tourist Hwy.
If you take the latter, when you get to the end, turn right to go to Sosua...
For bus service, take Caribe Tours from Santo Domingo to Sosua. Simple!
HB