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 available at the airport and at most hotels lobbies 24/7. The taxis are reliable and have fixed by taxi union prices.

When arriving, you'll walk out off the plane right outside using the wheeled stairs that connec tto the plane door.  You will be lined up on the tarmack and you must then wait to follow the airport personall into the terminal. Upon arrival, every tourist must buy a tourist card ($10 USD) before entering customs. Many tour operators have included this in your package or airfare and will have received the Tourist Card on the flight.  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. You may buy it online in advance and save time on arrival (just bring printed copy with you) - http://www.dgii.gov.do/tarjetaTuristi...

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. If you choose to let them take your luggage please sure to tip them, that is why they are there. 

If you don't have prior arranged transportation or a transfer included in your vacation package then you can easily just grab a waiting taxi to go to your resort. The rates are set. You will be taken directly to your resort. You can do the same in reverse and ask your resort to call you a taxi to return to the airport. they are available 24/7, reliable and hassle free.

You can as well buy a private airport-hotel-airport transfer from a lot of companies here are some: 

Dress comfortably for the airport.  As stated above, Terminal A it's an open air building and Terminal B is with strong air conditioners.  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 in Terminal A (only in the new Terminal B). You have to exit and enter the plane on the tarmac via portable stairway.

When departing, if your destination is the United States, you will need to first have your luggage screened by the "agricultural check" and have the appropriate stickers applied before you can check in. Once you're checked in, you'll go through security screening and then immigrations before you enter the departure "lounge" area.  Note that duty free is in two different areas - one as you walk in to the departure area to your right and there is another store in the air conditioned departure area which is to your left and down a flight of steps.  Tip: Grab an immigration form prior to entering the security line and fill it out there.   Duty free does not carry many of the Dominican rums and liquors and the prices of the Dominican alcohol in the other stores are outrageous so consider purchasing your Brugal and other Dominican brands BEFORE you get to the airport and pack it in your checked luggage. If you are transfering planes in the U.S. before your final destination, you will also need to go back through security and thus the 3-1-1 rule applies again. Any liquor or perfume you purchase in the Duty Free shops over 3-oz must be transferred into your checked bag when you clear Customs in the U.S.

Food options in the departure terminal are standard  American chain fare - Nathan's Hot Dogs, Wendy's, Baskin Robbins etc. and a Cinnabon's.  Be prepared to pay U.S. $9 or more for a Wendy's meal per person - about the same as other airports around the world. You could also load up with some food from your resort before you leave. A bottle of water costs U.S. $4 or so. Beer prices are approximately $5