Split to Havr to Dubrovnik transportation and length of stay

Are you planning to leave Zagreb the day you arrive? You can go to Plitvice by bus, then onward to Split by bus. If you wanted to make that slightly easier, you could look into a private transfer from Zagreb, but I don’t know what that would cost. Pltivice to Split is longer, and I would take the bus. Zagreb bus schedules available at https://www.akz.hr/. Get the schedule to Split from the tourist office at Plitvice there are hotels within the park near the bus stop. It’s that or rent a van, and while the drive is not difficult, I would not do it on arrival (unless flying within Europe and not overseas).
From Split to Hvar, check catamaran schedules at https://www.krilo.hr/ and https://www.jadrolinija.hr/. Some go direct to Hvar town, others to other towns. Stati Grad for example is linked to Hvar by bus. Hvar is tiny village, you don’t need transport for 1-2 days. You can take a boat ride or do an excursion via the tourist office.
Use the same websites for schedules onward to Dubrovnik.

In Croatia the buses are usually the quickest and most economical way to get around. A rental large enough for the six of you and your luggage will cost more than $1100 . Add parking and fuel to the cost and it vehicle rentals begin to fade as the optimal choice. The fact is you’ll wish you weren’t saddled with a vehicle once you arrive at Split and Dubrovnik. Taking the Krilo catamaran takes 75 minutes from Split to Hvar. The Krilo catamaran takes about 3 hours from Hvar Town to Dubrovnik. Make sure your ferry/catamaran’s destination is Hvar Town, because some go to Stari Grad. About anywhere in town is walkable as are places along the promenade north of the harbor. I stayed at the Villa Hosta Apartments near the Amfora Hotel last September and walking there with wheeled luggage from the ferry dock took all of 15 minutes. I would caution you though that in Hvar, the farther away the hotel is from the waterfront, the more likely it is uphill as the town was built into the hillside. There is absolutely no need for a car in Hvar Town ( or Dubrovnik, or Split if you have only two nights) With only 1-2 nights Hvar Town is about all you’ll have time for on the island.
The bus times for Zagreb to Plitvice
is 2.3 hours; Plitvice to Split is 4.5 hours. I would book the $50 flight from Dubrovnik’s Airport back to Zagreb because the 55 minute it takes beats the 7-10 hours the car or bus ride will take. (See Skyscanner.com for details)
Have a great Trip!

Bus is best from Zagreb to Split with stop in Plitvice. Bus transport is a great, easy, good value mode of transport.
Use ferries to enjoy a mode of transport utilized for eons in this area, by sea!
https://www.croatiatraveller.com/Transport/FerriesinCroatia.htm
Tip: water will still be chilled but pack some small inflatables for helping enjoy the swim experience. Inflatables cost a ton of money in Croatia. Split has a good public beach with easy access near Palace.
The experience of Havr is staying in Havr. No need for other ground transport as Havr is tiny, but vertical. Lots of cobblestone steps so wear comfortable well broken in walking shoes. Assign girls research on best clubs to party. Would not splurge on being near water in Harv as everything is near water in terms of walking time. Spend money on purchasing clothing, not junk tourist stuff, which will serve as great souvenir back home. "This old scarf? Thanks, it is lovely and I found it in Havr." makes for a wonderful wearable memory.
Take ferry to Dubrovik, again an excellent experience to relax and enjoy the coastline. Should not have an issue with obtaining tickets, but recommend reservations in order to gain maximum savings. Do online research via YouTube to see examples of the travel.
Dubrovnik is the most expensive of all Croatian towns. Take a deep dive on figuring out what you desire to spend money on. Fully research recommended guides before engaging their services. A lot of history to appreciate via use of great guide.

All this information is so helpful. Thanks to all the responses. Helps me understand much better.

Overall question. Anyone have a car/driver company they trust to use? We may need this for one day due to air/bus schedules not working out. Thank you!

You can check car rates on rentalcars,com. and autoeurope.com. Avec rentals is highly-rated.

We rented our last car in Croatia through booking.com. It was Sixt.

Looking for a car/driver service company instead of renting a car. I have found mydaytrip and also viator ( owned by trip advisor). Anyone have any experience with either company for private car/driver? Thank you!