Buses meet all car ferries and the bus ride between Vela Luka and Korcula town is an attractive way to get a glimpse of the island. The ride takes about an hour on comfortable buses and climbs through forested hills overlooking the sea.
From Hvar Island and Split you can also take the daily, year-round catamaran from Split to Hvar Town and then on to Korcula. See the schedule. In summer, there's an additional ferry that stops at Prigradica. Check the schedule.
The summer passenger ferry Split-Milna-Hvar-Korcula-Dubrovnik leaves in the morning from Split which makes it possible to do a day trip to Korcula from Split. See the schedule.
From Dubrovnik there are two summer only routes that allow a day trip from Dubrovnik to Korcula. There's a daily route that leaves every morning from Dubrovnik, going on to stop at Korcula, Hvar and beautiful Bol beach on Brac island before going on to Split. See the schedule.
Or, there's the catamaran route on the "Nona Ana" that runs from Dubrovnik to Korcula in the morning, stopping at Mljet and Sipan before heading back to Dubrovnik in the afternoon. It only runs in July and August though. See the schedule.
From Dubrovnik Airport
There's a private shuttle bus between Dubrovnik airport and popular destinations in Korcula and the Peljesac peninsula. From 30 April to 1 October (and possibly longer) four buses daily (only one on Sunday) will leave the airport for Vela luka (40 € /300 kn), Brna (40 € /300 kn), Lumbarda (34 € /255 kn), Korčula (32 € / 240 kn), Orebić (30 € / 225 kn), Trpanj (34 € / 255 kn), Prapratno (22 € / 165 kn), Slano (17 € / 125 kn) and Orašac (12 € / 90 kn). Prices include the boat trip. Reservations should be made at least 12 hours in advance online at Korkyra agency.
Buses to Korcula town meet most boats landing in Vela Luka and there are regular, if infrequent, bus links to other towns on the island.
As Korcula is an island, naturally, you'll need to take some ferries to transport your car but the following is a good route to take if you have a car but if the ferry schedules above are inconvenient: Drive to Orebic crossing the Peljesac Bridge and take the ferry shuttle from Orebic to Domince (a few km outside Korcula Town). If coming from Dubrovnik, you could simply drive across the Peljesac peninsula to the ferry shuttle at Orebic. It's a pretty drive.
If you're staying in Dubrovnik and wish to visit Korcula for the day, one practical solution is to take an organized tour to Korcula Island that gets you from Dubrovnik to Korcula, guides you around town, includes a wine tasting and leaves time for a swim.
All about Korcula in Croatia Traveller's Dalmatia: Split to Dubovnik