Forget the suggestion about the ferry. Only way to get there (unless you fly or charter a private boat) is by road and that road takes you to Santo Domingo first.
The suggestion to break it in two is a good one. You are talking about an approximate 500 km trip.
Would suggest you overnight in Colonial Santo Domingo, to get a feel for the old city that is quite a sight both during the day and the night. See listings for Santo Domingo hotels in
http://dr1.com/directories/index.shtml
Basically you will be going west to Santo Domingo and then north to Puerto Plata. I would bring a compass to help you keep these coordinates. Also, I would get a cell phone with a Dominican number -- these can be had for about US$50 including a calling card -- at Orange or Verizon shops.
There have been quite a few posts on this forum with directions. Please do a search. But basically, you need to go Punta Cana-Veron-Higuey-La Romana. Once you get to La Romana there is an expressway to go to Santo Domingo. I would stop in a gas station in La Romana and try to find someone who is driving that way that you can trail, so you do not get lost. Then you can skip the town of San Pedro de Macoris, and the expressway will put you into Boca Chica and Las Americas Expressway. This leads you to cross the city by way of the Juan Bosch bridge.
Here if you are going to the Colonial City do not take the overpass (expreso lane). Take the left lane that heads south into the colonial city.
If you have decided to do the trip in one go, then take the overpasses all the way through until you see a sign that says Dr. Delgado. Do not get off on that one, but the next. The next will take you to a small street that will lead you west to Leopoldo Navarro street. Then you take the Leopoldo Navarro street north to San Martin Avenue (left here at the stoplight) -- about a km. A slight swerve to the south will put you on John F. Kennedy Avenue. This will convert into the Autopista Duarte highway that will take you to Puerto Plata, via Santiago-Navarette.
We always recommend travelers to stop and stop. Buy a good map (sold at many gas stations) and ask and ask as you go from point to point on the map. Since you will have the map, show people where you are headed, and they will tell you what hte next stop is so you keep on mark. Dominicans are friendly and will help you.
Driving is an excellent way to get to know the country, but we recommend that you start early, and as recommended above, do not drive at night.