By ground transportation...
Go to Higuey and take the bus to El Seibo. In El Seibo you have the option of the bus to Miches or the one to Hato Mayor. I would prefer the one to Hato Mayor because the Miches road is too winding for my taste. You may need to overnight in Sabana de la Mar to catch the ferry the next day to Samana. Plan on traveling all day.
Another long option is to go to Santo Domingo (express bus from Bavaro) and then a Metro (7 and 3pm departure, $300) or Caribe Tours (7, 8:30, 10, 1:30, 2:30 adn 4pm departures, $280) to Samana. Each stretch takes about four hours, so about eight hours of bus time.