If you do not want to rent a car, your best choice is to take a taxi. This is expensive, though. The rate is US$85 or about RD$1,500 for the two hour trip. The advantage is that the taxi will drop you off right at your hotel. Best would be to get together with others who need taxi transportation on your flight and then split the taxi fare. If he makes separate stops in Punta Cana/Bavaro he will probably charge you extra, so negotiate that before taking the cab.
You could do it for about half, but you would have to have time. That is, you would take a taxi to La Romana (that taxi will charge you US$13 or about RD$230), and in La Romana you would catch the Sitrabapu express bus to Punta Cana for about RD$60 each. Also note that it is about half an hour from the airport to La Romana city, plus ironically, the bus will again pass right by the La Romana airport.
The downside is that when you get to Punta Cana, it could cost you about RD$150-RD$300 to get to your hotel. If you are staying at Barcelo Bavaro hotels, the bus will drop you off at the entrance to the hotel, and then you can catch a taxi for RD$60 into your hotel. Catalonia is also near the bus stop, but others may cost you the RD$150-RD$300 depending on how far away the hotel is from the bus stop.
