Hi, I will be traveling to Croatia with my husband and our two 20-something kids. We like a mix of city exploring and enjoying nature. We don't want to go somewhere new every day and be rushed. We will be in Split June 7-9, and Dubrovnik June 11-14. I was thinking we'd take a catamaran/ferry from Split to Dubrovnik but stay somewhere in between, on the way down. Should we do Hvar? Brac? Korcula? (I assume we should not try to squeeze in more than 1?) Would appreciate advice; thanks!

I think you are wise to consider only one stop.

I haven't been to Brac.

Hvar Town is much more upscale than Korcula; its harbor typically holds many yachts and there is often a lively bar scene in season. It has a medieval portion (as does Korcula Town), but the architecture in it's main tourist region has many Renaisance features. In contrast, Korcula Town's architecture is more typically Gothic / Medieval. Another option is Stari Grad, another town on Hvar Island. It's very laid back, on an quiet inlet rather than the sea, decidedly medieval.

These are very different places. I would encourage you and your family to consult a good guidebook or two -- the Rough Guide if you can find it -- to see what most appeals.

We will do it in a different way. We have booked a small ship cruise (7 nights) in September.
Itinerary is similar to this one: Antaris, Croatia - LiveAboard.com

I agree with kja that you are wise to consider only one stop. Of the 3 islands that you mention, we only visited Hvar as a daytrip from Split, and we loved Hvar. We enjoyed our walk up the hill to the fortress, following a zigzag path lined with pine trees and cacti, with spectacular views of the harbor and the town. Another highlight was our walk along the coastal path just west of the Old Town. The path leads past a cactus garden and green pine trees that contrast with the blue skies and blue water. Just gorgeous. There is a cafe along the way where we had drinks surrounded by beauty. I know that Hvar Town has a party reputation but it was quiet in September. I think June should be somewhat quiet, too. July and August are the busiest months.

Here is the link to my TR if you wish to read it.
Trip Report Croatia, Slovenia, and a day in Montenegro! Wow!! So Many Beautiful Places!! - Fodor's Travel Talk Forums

Do we need to rent a car to get around either of those islands?

No.

I haven't been, but I think the town of Omis on the mainland looks stunning! We're hoping to incorporate that into our itinerary when we go to Croatia this summer. It's about 30 minutes from Split.