Trying to get the most of southern Italy’s both coasts. Alberobello fascinates me. Does it make sense to fly in on the east coast for the Gargano Peninsula, Alberobello & Matera then somehow connect to the Amalfi coast & continue to Sicily? Are there other must see places on the east coast? If using a rental car can it be taken to Sicily & returned there?
Are there other must see places on the east coast?
Yes, but they are not in the area where you seem to have focused your attention. The Adriatic coast runs from Trieste to Santa Maria di Leuca. As a side note, if you talk about the East Coast in Italy, they think you are talking about LA.
There are buses from Matera to Salerno and ferries from Salerno to Sicily; but Salerno is not exactly on the way between Puglia and Sicily.
I think there is a Ryanair direct flight from Brindisi to Catania.
If using a rental car can it be taken to Sicily & returned there?
Yes, hundreds of tourists do it. Be aware that you'll pay more to return the car on an island and that you should really get an insurance without excess. You'll also have to rent from a Company that allows their cars to be loaded on ferries. Most do, but not all.
How much time do you have?
I would fly to Sicily and rent a different car while there.