You have a few different "Public Transportation" options when traveling in the DR.

Gua-guas are the mini vans jam packed with people, they will normally honk and call out to you, an inexpensive way to get somewhere but you have to fore-go any attachment you have to personal space.

Publicos are the sedan style cars with a taxi-like sign on the top.   With less people (but still crowded) these are the choice for Dominicans who need reliable transport to work and home.   Flag them down and if there is space they will stop to pick you up, but be ready to squeeze!

Motoconchos are the motorcycles that honk and call out to you every few moments.  These are the locals favorite for short, cheap trips. They can be very dangerous, have lots of accidents....

Motor Coaches such as Caribe Tours and Metro offer Bus service in large coaches to many parts of  country at low rates. They are very comfortable and considered safe, but make sure to bring something warm with you as airconditions in these buses are exteremly cold.

At all Dominican Republic Airports and eleswhere you can freely and easily catch a cab to your area hotel once you exit the airport, note that many resorts are very far from the airports. Take care in selecting which airport you fly in to. Taxi rates are prominently posted outside of customs at all airports.