Best way to go would be by plane (takes about an hour), or by taxi (will take you directly to your destination). But these are two costly options. The best way then would be by bus. The Metro or Caribe Tours bus that puts you into Santo Domingo, and then the Expreso Bavaro that will take you to Punta Cana. From there you take a local commuter bus or a taxi to your final destination. For more info on local transport, check out the Welcome Punta Cana guidebook at
Punta Cana - Dominican Republic