Travel help please :)

We're heading to Italy in September/October as a family of 4. This will be our 3rd Italy vacation but the first time in Puglia, Sicily and Malta. Trip is planned for 19 days. We don't like rental cars and prefer to make use of public transport/trains.

I'm struggling to figure out firstly- which order to plan the trip in (Rome-Malta-Sicily-Puglia-Rome or the other way around). And secondly- the best way (time & cost) to get from Puglia to Sicily(train or bus?) and then from Sicily to Malta (ferry or flights?).

We're thinking 7 days in Puglia, 6 days Sicily, 4 days Malta, 2 days in Rome on arrival and departure. We'll be back to Italy again so if you think we should remove one of the destinations as a seperate trip for another time, which one would you leave out? Thanks in advance :)