Of course, sailing there on your own sailboat of yacht is in a league of it's own. But, other than that:
1) Take a ferry from Orebić - the ferry ride is short, cheap and because of the configuration, almost always operating, uninterrupted by the winds. The drive to Orebić might not be the nicest one of your life in terms of road quality (although, that should drastically improve once the Pelješac bridge is finished), but the view and all that Plavac Mali wine on the Pelješac peninsula, which you'll traverse most of between Ston and Orebić, will make up for it. If you have time, go a bit further, from Orebić to the tip of the peninsula, because tourists rarely do that, and the views are spectacular, as well as small secluded beaches you can find along the way.
2) Take a ferry from Split - that's probably the most conservative option, although quite pricey for a car, and it will take you to Vela Luka, which is great if you're staying in the Western part of the island, but you'll have to drive for almost an hour to Korčula (not a big problem, of course, unless you've travelled from far away and are truly tired and even more so after the 3 hours on the ferry from Split). The ferry continues on to Lastovo, which is something you should consider for a day-or-two trip during your stay in Korčula
3) Take a catamaran from Split - this is probably the line with the most options during the day to get to Split, as both Jadrolinija and Krilo offer you several options a day. Make sure you check all of their timetables carefully, and purchase tickets either online or in their physical stores. And don't worry about the transfer from Vela Luka to Korčula or vice versa, there's regular bus service between the two you can count on, and it's really affordable.
4) Take a catamaran from Dubrovnik - If you're flying to the Dubrovnik Airport, don't worry, it won't be a problem for you to get to Korčula. Krilo will get you to Korčula from Dubrovnik (via Mljet) every day in the afternoon or evening (there are actually two catamarans, one going to Pomena and the other one to Sobra on Mljet, but both coming to Korčula), and during peak tourist season G&V Line will take you from Gruž harbour to Korčula four times a week in the morning (also stopping on Mljet and continuing to Lastovo)
5) Take a bus - there's quite a lot of options on how to get to and from Korčula by bus. There's a regular direct bus line every day from Dubrovnik to Korčula in the afternoon, taking just over 3 hours and going down Pelješac. There's also a regular bus line to Zagreb, going through Split, every day, leaving from Korčula in the evening and getting to the capital early in the morning (and vice versa, it leaves Zagreb in the evening and gets to Korčula at 6:35 am). From Split, Zagreb and Dubrovnik you can catch any type of transport to basically anywhere else. If you find yourself in Ploče, and you want to get to Korčula, remember that there is ferry from Ploče to Trpanj on Pelješac, and a quick bus ride from Trpanj to Orebić, and once you're in Orebić, it's not problem at all to get to Korčula.
