Getting to the Dominican Republic is very easy; you can get a flight from virtually anywhere in the world. There are regular flights from almost all major airlines and seasonal charter flights from most European and Canadian cities.
The main airports are:
(AZS) Samana, also known as "El Catey", located between the towns of Nagua and Samana on the north coast
(JBQ) "La Isabela" airport in, 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 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 (the country's 2nd largest city)
There is a charge of $10.00 for a tourist card on arrival. This must be paid in $US or Euros (Local currency, GBP, etc, will not be accepted. The $20.00 departure tax is included in the ticket price of most charter and some scheduled flights. Check your airline for details.
Another way to get to the Dominican Republic is by Ferry. There is a ferry that travels between Mayagüez, Puerto Rico and Santo Domingo. The journey takes about 12 hours, leaves Puerto Rico on Mondays, Wednesdays and Fridays at 8PM, and arrives in Dominican Republic at 8am the next morning.