Public transport in Dalmatia is cheap and efficient. The most common modes of moving about are buses and ferries. I've given a condensed summary of the most useful services below. See more on public transport in Dubrovnik on my How to get around Dubrovnik page.
Frequent buses (about 12 per day) run between Split and Dubrovnik, following the lovely coastal highway with the turquoise Adriatic Sea and the scattered Dalmatian islands in full view. The journey takes approximately four hours, and passes through a short coastal strip belonging to Bosnia Herzegovina – you'll have passport checks on the borders going in and coming out, with a short coffee stop in Neum. Long-distance buses in Croatia are operated by numerous private companies, each working on their own terms - prices and comfort vary slightly from company to company, though ticket prices are always reasonable.
On the islands, local buses are scheduled to coincide with ferry arrivals and departures, and connect the ports to most reasonably-sized outlying villages.
Split bus station (www.ak-split.hr; opposite the ferry port) is a 10-minute walk from the old town.
Dubrovnik bus station (www.libertasdubrovnik.hr; next to the ferry port) is a 25-minute walk from the Old Town (also served by local bus).
Croatia's rail network fizzles out at Split, so Dubrovnik is not connected to the rest of the country by train.
Ferries and catamarans
There are regular daily ferry and catamaran connections between Split and the islands of Brač, Hvar and Korčula. Ticket prices are very reasonable - the majority of services are run by the state-owned company Jadrolinija (www.jadrolinija.hr). Ferries carry cars and people, while catamarans are faster but only take foot passengers, and are more likely to be cancelled when the sea is rough.
Brač's main port is Supetar. Jadrolinija operate at least half a dozen ferries per day from Split to Supetar, plus a once-daily fast catamaran service from Split to Jelsa on Hvar, stopping at Bol on Brač's south coast en-route. The coastal towns of Supetar, Bol and Postira are connected by bus.
Stari Grad is Hvar's chief port, and is served by several Jadrolonija ferries from Split each day. In addition, Jadrolinija run a once-daily fast catamaran from Split to Lastovo, stopping at Hvar Town (on Hvar) and Vela Luka (on Korčula) en-route, and a once-daily fast catamaran from Split to Jelsa on Hvar, stopping at Bol on Brač on the way. Stari Grad, Hvar Town and Jelsa are all connected by bus.
Although Korčula is closer to Dubrovnik than Split, it is better connected to the latter by sea. However, you can reach Korčula from Dubrovnik by car, driving almost the entire length of Pelješac peninsular and then taking a 10-minute ferry ride, operated by Mediteranska Plovidba (www.medplov.hr), across the narrow channel from Orebić to Domince on the island (2km from Korčula Town). The same route is covered by once-daily bus from Dubrovnik. Split is connected to Korčula Town by the once-daily Krilo (www.krilo.hr) fast catamaran service, stopping at Hvar en-route.
In addition, Jadrolinija operate two services from Split to Vela Luka, on Korčula's west coast. There's a once-daily catamaran from Split to Lastovo, stopping at Hvar Town and Vela Luka en-route, and a once-daily ferry from Split to Lastovo, stopping just at Vela Luka. Korčula Town and Vela Luka are connected by bus.
Jadrolinija run several ferries daily from Dubrovnik to the nearby Elafiti islands of Koločep, Lopud and Šipan. In addition, G&V Line (www.gv-line.hr) operate the Nona Ana catamaran from Dubrovnik to the island of Mljet, stopping at Luka Šipanska on Šipan on the way. In July and August only, the route is extended to include Korčula Town.
As well as connecting the islands, Jadrolinija operates a twice-weekly overnight coastal service (with cabins available) running from Rijeka (in north Croatia) down to Dubrovnik, stopping at Split, Stari Grad (on Hvar) and Korčula Town en route. It's a slow but amusing way to travel between Split and Dubrovnik, if you have lots of time and enjoy boat journeys.