There is no airport on the island of Korcula, so you will need to fly to either Split or Dubrovnik and then make your way on to Korcula accordingly.

Ferries from Split to Korcula
There is a daily car ferry service from Split to Vela Luka on the western side of the island. Journey time is 3 and a half hours, and the service runs year-round. (Twice a day every day during summer; twice a day apart from Saturday and Sunday during off season.)

For something a bit more speedy, there's also a daily (year-round) catamaran service from Split to Vela Luka. Journey time is an hour and 45 minutes; the catamaran docks at Hvar Town on Hvar on the way.

During the summer months (June - September), Split and Korcula are also connected as part of the coastal car ferry route. This operates twice a week, and journey time is 6 hours.

All of the above ferry routes are operated by Jadrolinija.

Ferries from Dubrovnik to Korcula
The above-mentioned coastal car ferry also goes to/from Dubrovnik. Journey time is 4 hours and 10 minutes. Once again, this service only operates in summer when it runs twice a week.

During July & August, ferry company G & V Line run a daily catamaran service from Dubrovnik to Korcula (which also stops Sobra and Polace on the island of Mljet on the way) - it is their "Nona Ana" catamaran route. Journey time to Korcula is 2 and a half hours.

Orebic (Peljesac Peninsula) - Korcula/Domince Ferry
Korcula is situated very close to the mainland - only about 3.5km off the coast - and there are ferries from this part of the mainland (Orebic) to the island. This route is good to know if you're looking for more options in reaching the island from southern Dalmatia (e.g. you could get a bus from Dubrovnik to Orebic) - or perhaps you have your own car and want an hop on a car ferry to Korcula.

Ferry company Mediteranska Plovidba operate ferries from Orebic to Korcula Town year-round, with the service operating several times a day during off-season (apart from weekends) and more frequently during season. Journey time is 20 minutes.

Jadrolinija also operate a car ferry service from Orebic to Domince, which is a suburb on the edge of Korcula Town. From Domince to the main part of Korcula it's about a half hour walk, or you could get a local bus or taxi.

This ferry service is very, very frequent - there's almost hourly ferries during the day, even in off season. Journey time is about half an hour.

Bus from Dubrovnik to Korcula
There is a daily bus operating from Dubrovnik to the island of Korcula - using the above ferry route. This bus route is run by the company Autotrans - you can look at the timetable on their website.