Dubrovnik and Split

You can get from Dubrovnik to Split by bus (4h 30m) or ferry (5h 30m).

We recommend the ferry as water travel has been the predominant mode of travel for centuries. Doing so provides wonder vistas, experience of a different mode fo travel and the opportunity (should you choose) to stop at another destination to add to your experience. Ferry travel is a clean, commonly used, dependable experience.

A couple of recommendations.........

Split three days
Your pick of an island destination for two days
Dubrovnik three days

Seek BNB's at each destination.
In Split we stayed on the west edge of old town, use Konoba Fetivi restaurant location to locate the area, in a wonderful BNB. This is a wonderful quaint area with ample easy access to old town.

For an alternative trip consider taking a seven day small ship cruise which will carry you from Split to Dubrovnik. This is my trip report on the experience:
https://community.ricksteves.com/travel-forum/trip-reports/a-small-ship-cruise-in-croatia-october-2019

An a minimum examine the ports described in the report to help determine your interest level in where to visit.

Hope you are flying "open jaw" into Split and out of Dubrovnik as it will save time.

For Dubrovnik, you need to decide whether to stay at the beach or in the center of city. In Fall 2019, there was a bus that went from the beach to the center.

While in both cities we stay in the center. We like being able to walk out side and be in the middle of the action.
Split - only visited once for 3 nights. I like to have a short orientation tour of each city we visit to get lay of the land and some basic information. We used Pero Ugarkovic, whom we found through toursbylocals.com. When not leading tours, he is a marine biologist and grew up in Split. PM me if you want his personal email for a tour. We also did a day trip to Trogir, Salina, and Klis on a HOHO bus tour. But you can go to Trogir on your own. My daughter and her husband did a day at the beach snorkeling. This is a city with Roman history and amazing architecture. There are many museums and you can climb to the top of the tower by the Cathedral of St. Domnius. Not too many steps. We stayed at the Hotel Peristil, great location, nice rooms and breakfast, no elevator.
Dubrovnik - have been here numerous times and only stay in the old city. This city is medieval and very different from Split. Also take an afternoon tour because they explain about the walls and the history of the city. Again, there are museums spread about the city and many churches. Just walking up and down some of the streets will give you a workout. The nearest beach, Banje Beach, is a short walk out the Ploce Gate but go early (we go at 7 am) it gets very crowded and is not too big. Definitely take the morning ferry to Lokrum. Here is your best hiking. The entire island is a national park. The ferry leaves right from the pier and takes about 15 minutes. If you want to swim there for the day, take the first ferry and follow all the elderly people, that’s where they are headed. There is only one cafe on the island so you might want to bring some snacks. We also did the dinner cruise on the Karaka, twice. It is a lovely way to spend an evening. There is kayaking but we haven’t gone. We usually stay at Pavisa Apartments (up about 80+ street steps) or Bed & Breakfast Andio (off the Stradun but across from a pub). pavisa is in Rick’s book and our favorite. There is a tiny kitchenette in each room so you can make coffee or tea and cook if you want, also small fridge. We usually came back from the beach, showered then had scrambled eggs each morning. Andio is through booking.com and provides breakfast. Our favorite restaurant is Konoba Kamenice, outdoor seating only, which is located on the main square.
Once you start researching and googling the cities, so many sights to see and things to do will pop up. It all depends on your interests. Each time we visit we find something new to do - we’re going this September. Forgot, we go by ferry between cities, Krilo.

Another (very scenic) option: spend just 2 nights in Split, then take a morning ferry to the island of Vis. Spend 3 nights there, enjoying the hiking & scenery. Be sure to visit the lovely village of Komiza and, if it’s a sunny day, take a boat trip to the Blue Hole. After 3 nights, take the early morning ferry back to Split, walk across the street to the bus station and take a bus to Dubrovnik. Instead of staying in Dubrovnik, though, take another bus about 20 minutes south to the lovely village of Cavtat and stay there. Depending on Covid restrictions you may be able to do a day trip to Montenegro by rental car or bus tour. Or you can take a boat tour of Croatian islands from Cavtat. You can make a day trip into Dubrovnik by Bus or boat. When you’re ready to fly home, the Dubrovnik airport is just a few minutes outside of Cavtat.

There are hundreds of options, so I am going to offer general advice to help you narrow it down.
First, read about Dubrovnik and Split and determine how much time you want in each. You could easily spend four days in each, but I think you should tailor it to your interests. Once that is decided, you will know how much time you have available in the middle if you want to stay somewhere smaller, closer to hiking, on an island, etc.
You could take a bus down the coast, stay at Makarska in between, hike at Biokovo, or you could take a boat and stop on an island en route. Ferry operators are Jadrolinija and Krilo. You have time for one island at least.
August is high season, so book early. I would avoid Hvar town in August (though your daughter might like it).

Thank you all for your recommendations !!! I am very excited to start narrowing it down and making some reservations! Thanks again!