The Punta Cana International Airport (PUJ) is privately owned by the same group which owns Punta Cana, as well as the PUNTACANA Resort & Club and Tortuga Bay. If you're staying at either of those two resorts, you'll get VIP treatment at the airport. Someone will escort you through customs, you'll skip all the lines, and there's even a private lounge for you to wait in when you're on your way out. It's the closest thing we've ever felt to being famous. There is also the La Romana Airport (LRM) about 1.5 hours away, or you can fly into the Las Americas International Airport (SDQ) in Santo Domingo and take a four-hour commuter bus ride to Punta Cana for about $8. Be sure and bring $10 with you if you're flying into PUJ because you'll be required to purchase a $10 Tourist Card before entering customs unless you have a Dominican passport.
If you're already in the Caribbean, it is possible to catch a ferry to Punta Cana or Santo Domingo from some major cities such as San Juan, Puerto Rico. Additionally, the PUNTACANA resort features a full-service marina with mooring for 43 vessels up to 70 feet long.
Most of the major rental services are in Punta Cana and doing quite well. Check ahead to see if there's a rental agency at your hotel in case you don't want to deal with it at the airport.
There are two main taxi companies which serve Punta Cana, Taxi Turistico Beron 809-466-1133 and Taxi Bavaro-Punta Cana (Siutratural) 809-221-2741, 809-552-0617.
For the adventurous and patient, you can use the local minibus system in Punta Cana very cheaply. The buses are called "guaguas," and they travel regular routes at irregular times. To catch a guagua, you head to the nearest stop and say a little prayer—and be sure to wave your arms when you see one coming. They'll let you off wherever you want but remember you might have to wait a while to get a bus back.