If arriving by plane from almost every European or international airport in the season you have the seasonal and charter flights to Dalmatia. Direct charter flights to the largest touristic destinations: Split, Dubrovnik, Zadar, Brac, part of regular flights are via Zagreb airport.
For entry and driving through Dalmatia, Croatia you need valid driving license and a insurance green card. During summer months traffic at the border entrances is more densely, especially on weekends, what can cause short hold ups. Toll is charged on the highways and some bridges.
Regural international bus lines conect Dalmatia, Croatia with Austria, Italia, Hungary, France, Germany, Bosna and Herzegovina. During the season there are much more international bus lines to Dalmatia.
This kind of transportation is the cheapest if you come to Dalmatia from eastern Europe. Croatia has also quality developed national bus lines net, that reaches to the smallest places on the Dalmatian coast.
You can travel by train to Split, Zadar and Sibenik via Zagreb, Dubrovnik is not connected by rail.
The islands can be reached by ferry or catamaran from Split, Dubrovnik, Zadar, Sibenik, Makarska, Drvenik ...
On the islands, local buses are scheduled to coincide with ferry arrivals and departures, and connect the ports to most reasonably-sized outlying villages.
Of course there are other alternative means of transportation as well as rent-a-car, taxi, shuttle, donkey...