Motorcycles on the highways, given driving habits of Dominicans and road conditions, is not the best way to see the country.
Cris Colon's suggestion about going to Santiago for a day is a good one. Caribe Tour has a bus leaving Puerto Plata at half past the hour all day long. To Santiago takes less than 1.5 hours and you will see a lot of country enroute. It will cost you about RD$50/person each way. When you get to the Santiago terminal, take a taxi from there to the city. The taxi will charge you about RD$50. When you get ready to return, take another taxi back to the Caribe Tour terminal. Santiago is the 2nd largest city, and it is not a tourist city. You will see very few people that might be tourists on the busy streets.
Another day you can take a guagua or public car (publico) to Sosua. The public cars have a red sign on the roof that says Sosua-Puerto Plata. They are passing the central park all the time. The guaguas also leave from the central park. To Sosua, the public cars cost RD$15/person and the guaguas cost RD$10/person. I usually take the cars because they get to Sosua faster. When you get ready to go back, take another car or guagua to Puerto Plata. There is a very nice beach in Sosua, so bring your suits.
Another day, or same day that you go to Sosua, you can go to Cabarette. It is a very nice town and is one of the best windsurfing places in the world. Nice beach there, also. You can take a guagua from the central park in Puerto Plata to Cabarete, passing by Sosua. Cabarete is just a few miles past Sosua. Or you can take a public car from Puerto Plata to Sosua, then switch to another public car (both operate from same location) to get to Cabarete. From Sosua to Cabarete is RD$7/person each way.
Take those trips and you will see a lot of country without spending much money.