Split to the islands and then Dubrovnik

There is a daily ferry from Split to Dubrovnik which is not expensive. It stops at 2 islands. I believe that you can get off and get on. I think Korcula and Hvar. But the ferry is online and the schedule is listed. You do need to ensure that the ferry will be running - it does stop in the fall - just not sure when. We've been to Croatia 6 times since 2011, but haven't spent time on the islands. As to sand beaches, there are few in Croatia - one in Split, one in Cavtat, a few other places - most beaches are pebble or rock.

I prefer longer stays, so I would do 2 nights in Split & 3 nights on an island. You might consider the tiny island of Vis, reached via ferry from Split. There’s an early morning ferry back to Split that connects with a bus to Dubrovnik. Easy logistics. No need for advance tickets.

Quiet little Vis has no wineries but lots of good food & scenic wandering. We love the tiny fishing village of Komiza, from which (on a sunny day) you can take a boat tour to the beautiful Blue Cave on Bisevo. Vis has a big yacht harbor. We loved sitting by the water at the end of the day with a glass of wine, watching all the boats come in.

Visit Trogir for half a day on the way to Split where you can drop the car.

I would start with a couple of nights in Split, which gives you one full day there, which is enough for the highlights. I would then take the ferry to Stari Grad on Hvar (you don’t need advance tickets for this) and stay either here or Jelsa or Vrboska as these places have a great selection of cafes and restaurants and there is great walking on Stari Grad Plain, a UNESCO World Heritage site. Hvar Town will be quieter by late September, but there will still be some partygoers left. I prefer the north of the island.

You will need to buy tickets in advance for the Hvar - Dubrovnik leg, as this is a daily catamaran, not a ferry - Krilo.hr and runs each morning.

Please be aware that many places on the islands close at the end of September and start winding down from mid September onwards, so it will feel quiet.

I really hope you get to make this journey as Croatia in September is a great time to visit.
If you can arrive in Split by mid-morning then I suggest you spend one night and take a late afternoon ferry to an island.
Stay centric to Old town Split and the defined amount of time is sufficient to take the flavor of this destination.
My recommendation is to take two nights in Hvar and Korcula while recognizing these will be slower relaxing times to explore these destinations and surrounding area. I advocate slowing down in these two villages as they offer romance, history and the opportunity to enjoy some wonderful strolls. We enjoyed these destinations and given the effort required to get to these destinations I advocate slowing down and enjoying the stay.
Safe travels and be well!

I would not recommend more than 1 day in Split. It is a nice town but except for the small part of waterfront (Croatian-"riva") (2-3 hundred meters) near the city harbor and around Diocletian's palace it does not have that mediterranean feeling that many other smaller towns in Dalmatia have. You can walk through the Split old town in 1-2 hours. The rest is basically an industrial town with 250 thousand inhabitants with all things that go with that, like pollution of air and see, criminal etc. If you really want to feel life in Dalmatia go to Makarska or to any of small villages around it (Makarska riviera) it is more natural greener, sea is unpolluted etc. I think that Makarska has a ferry connection to island Brac. Split is not good even for a base for day trips because scenery of land and see around it (storehouses, cement factories etc) is not even close to that of around Makarska and further south to Dubrovnik

I am going to push back on the last post painting a bleak image of Spilt as we based there for five days and enjoyed a wonderful experience. The area of the palace is small and can be "visited" within a day, but the variety of interesting actives and people to be engaged outside the Palace offer opportunities to create wonderful travel memories. We like walking outside of typical tourist paths to learn how locals live and absorb some of their experiences. In Split we created a wide circular path going from the Palace through neighborhoods, parks, schools, museums and to the football stadium then crossing to the back bay and finding several shopping centers and a mall. We did not encounter a criminal element, but did find locales helpful in guiding us to a neighborhood market, a shortcut along a flowered path to the mall and places to stop rest and be offered refreshment. We completed our walk through a busy Main Street and stopped at various local retail shops and a pawn shop where the owner delighted in finding tourist coming into his business. He is a wonderful kind Croatian who shared many experiences and loved the fun bartering we did on purchasing a few local vintage items. We then found a clothing resale shop which offered another opportunity for engagement and where my wife literally traded the shirt off of my back for some scarves, a beautiful skirt and a old shirt for me (I was wearing a solid med blue quick drying Columbia brand travel shirt the owner insisted upon having...........plus I received a free drink at the next door bar.................), all creating wonderful memories. We ended the day dipping our tired feet into the water at the beach found just beyond the main port and enjoyed seeing children play, old folks playing cards and young men trying to impress pretty girls. Split is full of life and wonderful folks all waiting to be found beyond the gates of the Palace.

Adding to the previous post, there is a fabulous Archeological Museum in Split. We spent an afternoon there. Croatia was called Illyria during Roman times, and was a key part of the empire, with Diocletian and 18 other emperors from this area. So there is a lot of Roman history in Split.

Many thanks to everyone for your suggestions! @Marbleskies, those types of experiences are exactly what we seek when traveling; how fortunate you were to have that!