There's no train connection with Punta Cana Airport.
Car and bus
Punta Cana Airport is situated on the east coast of the Dominican Republic, in between the beaches and holiday resorts. Coming from the north you can reach the airport in your rental car via the Cap Cana freeway, from the south, you take the Boulevard Hotelero. Both roads connect with the Carretera Aeropuerto to the airport.
Buses appear to be running from the airport to destinations on the coast, but they run very infrequently.
Car rental Punta Cana Airport
You can find counters of car rental companies in the Arrivals Hall. Several rental companies have a presence at Punta Cana Airport, including international ones such as Avis, Budget and Hertz, but also some local providers. You can also book your rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Most people arriving at Punta Cana Airport have organized their transport via their hotel. If you haven't done so you can find taxis outside the Arrivals Hall. There are fixed rates for various destinations and most taxi drivers have a copy of this official price list. You can pay taxi drivers in the local currency, the peso, but also in US$ or €.
There's a parking lot outside the terminal, but only for short-term parking. We strongly advise against leaving your rental car here at night.