Getting around Croatia

Are any flights booked? What month? (If high summer, I'd do NO sightseeing and purely enjoy nature.)
Dubrovnik and Split are about 4 hours apart by bus, also linked by catamaran. I guess you could stay somewhere in between, but it could be easier to stay in both, stopping at Hvar in between.
Unless you are flying in or out of Zagreb, which would make a stopover possible, I would skip Plitvice and maybe visit Krka, which is in within day trip distance of Split.
Spend some time on the Krilo and Jadrolinija websites to get a sense of the catamaran and ferry routes, as that will make your planning much easier. You can also identify the ports on Google Maps by zooming in on the dotted lines.

There are buses, but we used Octopus Transfers for our land transfers, and they were great. https://www.octopustransferscroatia.com/ For transferring to/from Hvar, we used the ferry (catamaran).

Hi. I went on a trip to all those places in 2016, when we were in our late 50s, but without kids. (I have travelled with a number of teenaged kids to other places in Europe).

We travelled (after a stay in Slovenia) from Plitvice Lakes to Split to Hvar to Korcula to Dubrovnik (and then on to Mostar and Sarajevo in Bosnia). We had a rental car which we dropped off in Split, and took ferries to Hvar and Korcula, and then a van from Korcula to Dubrovnik.

You should try to see Plitvice Lakes; it was one of the most memorable stops on the trip. I would recommend staying nearby the night before, and being first in line at the Park when it opens. We spent the whole day, and felt it was worth a whole day.

It does involve a lot of walking though. Not sure how your 6 year old might handle that.

Getting there requires a car. If you don't want to drive, I would be very surprised if you couldn't arrange for transportation through a travel agency or excursion service of some kind from Split.

Split's big feature is Diocletian's palace. Its a huge complex; we actually stayed at a place located inside it.

I remember the ferry from Split to Hvar being quite crowded.

I am not sure Hvar would be a great town for your family. It's kind of known as a party town for college age kids.

I do remember attending a music performance that seemed mostly intended for the locals, and spending a pleasant day walking through the town and along the beach.

We stayed at a place that advertised itself as having "The Best View in Hvar." It probably did. But of course what that meant was it was at the top of a little hill at the edge of town. Which meant long climbs up and down steps. Not sure how your kids would handle that. With your family, I would try to find something in town, or just outside along the water.

If there were one stop I would consider cutting with your group, it would be Hvar.

Korcula is a bit sleepier town, and might be the better island stop alternative for your family. We enjoyed exploring it.

We took a travel agency van to get from Korcula to Dubrovnik. I believe the van service was recommended in Rick Steves' book.

I don't remember a ferry segment to the trip, but there must have been a short one. We had an interesting incident where the van's side mirror clipped the side mirror of a car coming the opposite direction; it was definitely the other car's fault.

In Dubrovnik we stayed at a Rick Steves recommended place in the old town; the place and its proprietors were featured in his TV show on Dubrovnik. It worked well.

Good luck!

Thanks all! This is very helpful information.

I count 7 people in your group. Small car, 4 adults, 3 children, long drives. Two lane roads, two lanes and winding. Consider that. Plus luggage.
One week for your sights does not seem reasonable. Perhaps day tour out of Split for Plitvice, but be aware of the very crowded nature of Plitvice. Very crowded. And when the tour bus arrives you would be in the thick of it.