Although the Island is large, the only land border that the Dominican Republic shares is Haiti. Only 4 borders crossings exist, including one between Dajabón and Ouanaminthe, another west near Elías Piña, another one near Jimaní, and another one between Pedernales and Anse-à-Pitres. Some of those crossings can be open 08:00-18:00.
The main airports in alphabetical order are:
AZS Samana, also known as "El Catey", located between the towns of Nagua and Samana on the north coast.
EPS Samana, also known as "Aeropuerto Internacional Arroyo Barril" between Sanchez and Samaná
JBQ "La Isabela" airport in Santo Domingo, mainly for domestic flights but also receives some flights from other Caribbean islands
LRM La Romana on the south east coast
POP Puerto Plata, also known as "Gregorio Luperon" on the north coast
PUJ Punta Cana International Airport in the east, the busiest in the country
SDQ Santo Domingo, also known as "Las Americas" on the south coast close to the capital city Santo Domingo
STI Santiago also known as "Cibao International" in Santiago de los Caballeros the country's 2nd largest city.
COZ Constanza, a domestic airport to all dominican destinations.
BRX Barahona, aslo known as "Aeropuerto Internacional María Montez" this airpot was reopened during the earthquake in Haiti, in order to bring the primary aid to the haitians.
CBJ Cabo Rojo, Pedernales, only for domestic use, located near Cabo Rojo port facility.
You can get flights from Europe at least via Madrid MAD, Paris CDG and Munich MUC. From the US, you can fly from New York, Ft. Lauderdale, Miami, Philadelphia, San Juan, Atlanta or Charlotte. Most European and Canadian cities have charter flight connections, which operate seasonally.
You will be charged USD10 for a tourist card. This can be purchased on arrival or in advance through the government's web portal, which delivers the tourist cards via PDF download immediately. This must be paid in US dollars or euros. Local currency, sterling, or other currencies will not be accepted. A departure tax of USD20 cash is payable on most charter and some scheduled flights. If you are flying on a US carrier, the departure tax is always included in the taxes when you purchased your ticket, so you will not have to pay anything when leaving.
Taxi fares to nearby hotels are posted just outside the airports. From the airport to Santo Domingo Ciudad Colonial it's about USD40. There are no hotel "courtesy shuttles" at airports in the Dominican Republic.
At the airport, you can change your US dollars and euros in Dominican Pesos. Note that you may not be able to exchange back local money to US dollars and euros, so do it before leaving.
There is a ferry that travels between Mayagüez in Puerto Rico and San Juan in the Dominican Republic. The website says the journey takes 12 hours, leaves Puerto Rico on Mondays, Wednesdays and Fridays at 20:00, and arrives in the Dominican Republic at 08:00 the next morning. For prices and bookings, visit the Ferries Del Caribe English language page.