If you will be staying at a resort in Punta Cana, you should look into changing your flight arrangements so you can land at the Punta Cana Airport, which is less than half an hour from most area hotels.
If you cannot change these, your best option would be to take an Air Santo Domingo flight from Puerto Plata International to Punta Cana International, but your flight would have to coincide with their twice-a-day flights. They have flights at 11:15 am and 6 pm. It will cost you RD$700 per person, one way. It's probably the best way to go. You can buy the flight upon arrival, there usually is space. Tel. 809 683-8020
Another option is to spend two or three days visiting the North Coast, and then another day in Santo Domingo visiting the capital city and then head for Punta Cana.
Or head to Puerto Plata city from the airport (about 20 minutes) and then take a Metro Bus to Santo Domingo (the airport taxi will cost more than the bus to Santo Domingo which is about US$6 for the four hour trip). They leave several times a day. Once in Santo Domingo, a taxi from Santo Domingo to Punta Cana will cost about RD$1,500, one way. You can also fly to Punta Cana (Air Santo Domingo) for RD$500 one way per person with departures at 8:15, 10:30 am and 5 pm from Herrera Airport in Santo Domingo.
You could rent a car for that money and make the drive. You will surely get lost on the way, but if you ask and ask and ask, friendly Dominicans on the way will set you straight. Note that it's a four hour trip by road to Santo Domingo, and then another four hour trip to Punta Cana.
Unfortunately, there are no deluxe bus companies making the trip to Punta Cana. You would have to take a minibus to Higuey and then another bus to Punta Cana. This minibus-connection (Santo Domingo-Punta Cana) will cost you less than RD$100 per person (about US$7 per person).
Conclusion, the cheapest way is take a minibus to Puerto Plata from outside the airport, then a Metro to Santo Domingo, then a taxi to Parque Enriquillo to take the bus to Higuey, and then a bus to Punta Cana.
The most comfortable and carefree way is to take the Air Santo Domingo connection into Punta Cana, taking from there a taxi to your hotel.
Half way options are the Metro Bus, taxis and renting a car.