The "easiest" way is to go to Puerto Plata, Navarrete and Santiago, and follow signs to Santo Domingo. Once you are on the highway, about 10-15 minutes after you pass the big Sirena store, 8-10 minutes after you pass the last traffic light at Leche Rica, you will see big signs indicating the turn-off for the airport.
The other way through Cabarete, and Jamao, comes out in Moca. If you go right towards Santiago, when you get to Licey, turn left to the airport
If your go straight, you come to the Ramon Caceres road that leads to the Duarte Hwy...go right there to the airport entrance, amybe 5 minutes.
Good luck.
HB