car ferry from Split to Hvar to Korcula

I’ve only been on the Krilo Catamaran which doesn’t transport cars. With a car, you need to take a ferry from Split to Stari Grad on Hvar, and from Stari Grad to Vela Luka on Korcula. www.CroatiaFerries.com

Is there a reason you need to keep the car after you get to Split? If you can drop it off there, you can easily enjoy the islands and Dubrovnik without worrying about parking and driving in traffic.

If you do need the car, the link Kenko provided will give you all the info you need to find the right ferries. You'll just need to go through all the options to find the route and service that fits your schedule. For each ferry service, it'll tell you whether they accept cars.

• Split-Hvar ferries: https://www.croatiaferries.com/split-hvar-ferry.htm
• Hvar-Korcula ferries: https://www.croatiaferries.com/korcula-hvar-ferry.htm

As Lane wrote, you’l actually find a car more of a hindrance than a help if you’re planning on staying within the old walls of Dubrovnik, in Korcula Town and in Hvar Town. Both within and just outside of these locations, parking is not available or very problematic.
Taking the ferries/catamarans is much more enjoyable.

We're planning on visiting some wineries on the islands (nothing confirmed yet). Car seemed the best, most flexible mode of transportation. If we don't take our car to Hvar, is there parking in Split where we can leave the car for the day?

You would have to take the Jadrolinija ferry with a car and believe me, it is a hassle. When in Hvar or Korčula there are plenty of taxis or drivers willing to drive you around. I know on Korčula you could rent bikes to ride to the wineries, of course, if you plan on imbibing, I would hire a driver.