Florence to Puglia

Since currently there are no non stop flights from Florence airport to either Bari or Brindisi, the best option is to take the high speed train (many trains require a change in Rome). Going to Pisa or Bologna to catch a flight will not save you much time (maybe 4 hrs vs. 6.5 with the train), after everything is said and done (airplane travel requires security procedures before boarding that the train doesn't). The train is much more comfortable also.
Train info at the two train operators' websites:
www.trenitalia.com
www.italotreno.it
Use the following station names:
From: FIRENZE SANTA MARIA NOVELLA
To: BARI CENTRALE
Once in Bari, I recommend to rent a car. There are multiple car rental offices near the station. Many nice locations are more easily reached by car.
For rental cars I recommend:
www.autoeurope.com

Would you be interested in Visiting Matera? I know it's not in Puglia, but it's on the way.

Trenitalia sells a Train&Bus combo ticket from Florence via Salerno.

The big question is where do you want to go in Puglia? Bari? Lecce?

We flew Rome to Bari, fast and inexpensive. Of course, you’d have to get to Rome FCO.
Puglia is wonderful! We based in on the eater in Polignano a Mare. We went to Matera one day too.

We are open to suggestions of where to go in Puglia. We are also not set on Puglia and welcome any other suggestions as well - ideally would prefer to take the train and be under 6 hours. In the past we have visited Siracusa, Salerno, Amalfi Coast, Rome, Venice, and Florence. Cinque Terre is an option as well (although my daughter has already been there twice so last resort!).

How many nights were you thinking of visiting Puglia? Is renting a car out of the question? Do you and your daughter like art and architecture? Food? Scenery? Staying in a town or out in the country? I can definitely point you to places in Puglia once I have a better idea of what interests you both about the region.

To partly answer your other question, we have found the two easiest regions to visit using only public transportation (mostly trains) are the Veneto and Emilia-Romagna.

As a very general guideline, public transit is a bit thinner in the south than in the north simply due to lower population density. I was going to suggest Abruzzo (four hours Florence to Pescara if you pick the fast trains, up to nine hours if you don’t), but you won’t see much away from the coastline without a car. Up and down the coast there are trains, inland has limited regional train lines and buses that run infrequently, aligned with work and school hours. This is also true in Puglia - trains along the coast, some bus service inland. It’s definitely doable in both places but will require more advance planning than just heading to the train station like you might up north.

April isn’t beach season in either region but there are plenty of things to see in the beautiful little towns. In Abruzzo, bigger towns with train stations include Vasto, Lanciano, Chieti in the south, Atri, Teramo in the north, L’Aquila to the west, Pescara of course on the coast. If you do get a car, the little mountain towns such as Guardiagrele, Roccamorice, Caramanico Terme, or further west, Tagliacozzo, Santo Stefano di Sessanio would be interesting.

I’ve only been to Puglia once and enjoyed the coast in and around Monopoli. I found the interior a bit bleak and desert-like, though Alberobello was a joy. I’d like to see Ostuni.

In general, I applaud your idea to get away from the busier spots and see some of the rest of Italy. Driving, at least in Abruzzo, is not difficult as the roads are well maintained and well marked, with significantly less traffic than the north (I’m currently living near Milan and leaving the car parked as much as possible due to the crazy traffic.) The roads in Puglia were a bit rougher, especially around Bari. But a road trip would be easier and more fun than coordinating limited transit schedules, IMO.

April is a beautiful month weather-wise, and I’m sure you will enjoy whatever trip you plan.