The fastest way to get to Dubrovnik is by plane, the flight from Zagreb takes only about 50 min. Various airline companies have air-routes towards Dubrovnik, there are several flights per day to Dubrovnik. Croatia Airlines is the largest croatian airline company, and on their web site you can find all the departure and arrival information about the airplanes landing at the Dubrovnik Airport.
In order to arrive to Dubrovnik by car, using the croatian road-network, the easiest way is along the Adriatic highway, though there are alternative ways that are recommended during the summer season in order to avoid traffic jam and delays on the highway. To get more information about croatian roads, check out the Croatian Roads website
Dubrovnik is connected to almost every other location through the Zagreb Bus Terminal. One bus or coach comes from that direction almost every hour. To get more detailed information about departure and arrival of the busses and coaches from all directions, check out the Zagreb Main Bus Terminal website
If you have time and want to turn your trip into enjoyment, choose a ship as means of transport. The trip takes longer but then you get a chance to see the adriatic coast and enjoy in the beauties of the islands and the shore. Fast ship lines are mostly used by guests from the neighbouring country of Italy. For more information about ship lines, visit the Jadrolinija (Adria-line) website