Punta Cana is such a popular tourist destination that it has its own (privately owned) airport, which is located about 10 to 40 minutes away from most of the area resorts.  The airport itself is considered something to see, as it features a thatched roof and open-air design.  As it is the closest to the resorts, this is probably the best airport to fly into, and most major airlines fly there.  You may want to pre-book a transfer from the airport to your hotel, although there are taxis (tourist taxis cost nearly twice as much as the smaller city taxis).

A good second choice in terms of airports is  La Romana Airport, which is about one hour away from most resorts.  Cruise ships tend to dock at the larger ports. 

Upon arrival, every tourist must buy a tourist card ($10US) before entering customs.  It is said that the immigration officials only care about the entry fee and not whether you fill out the card, but the card is very simple and takes just a couple minutes to fill out. 

Dress comfortably for the airport.  As stated above, it's an open air building.   Until you are all checked in you will be without benefit of air conditioning.  (Unless you are one of the lucky ones who can locate and get seated in the COLD air conditioned lounge, at the food court to the right of the Wendy's.) The lines are not dreadfully long, but it could take an hour to get completely checked in.   Keep all your identifications handy.  You will have to present them at different locations up to 7 times.    Also, there is no indoor gateway.  You have to exit and enter the plane on the tarmac via portable stairway.

You may be approached by guys that want to cart your bags to your taxi. Make sure to have people in charge of certain bags so none are scattered while they grab your luggage.