Wow, Harold. Just read your trip report--very accurate & gives a good sense of Sicily, even though when we were there in 2010 we only stayed on the eastern coast, from Catania to Taormina, and the medieval villages of Savoca, Forza D'Agro and Castelmola. Yes, Sicilians like to say 'this isn't Italy', but of course, it is. It's just a little different. And quite wonderful.
Back to business. Let's say, KLT, you're flying in & out of Rome, and have 14 nights to work with. Fly Rome to Catania, and start from there. I have not driven in Italy at all, but if I were, Sicily would not be my first choice of places to drive, no way. We took private tours up, up into the mountains, towards Etna, also to see the film locations of the Godfather movies, which was fascinating. And had we driven, even with GPS, I'd have been a nervous wreck. Hairpin turns & cliffs scarier than anything in the Berkshires of Massachusetts, Blue Ridge Mtns of Tennessee or Yosemite Nat'l Park in California. To see all of Sicily, in my estimation it would take much more than a week to do it right, but 5-6 days is much better than not going there.
Thus, I would obtain private transport from Catania airport and base in Taormina, as we did. Explore the history and view the spectacular vistas while taking private daytrips, with tour companies at reasonable prices, to Mt. Etna, Palermo, etc. Fantastic dinners at night, enjoy my favorite cuisine in Italy. Then, even though it would be wonderful to take the ferry across the Straits of Messina to the mainland and take a bus to the caves of Matera in Basilicata for a day or two (hopefully my next trip), it's probably more efficient to fly Catania (or Palermo) to Naples, and chill out on the Amalfi Coast either in Sorrento or Ravello for a few days. In that time, it's easy to visit Naples, Pompei, even Paestum south of Salerno. BTW, we stayed 5 nights in Salerno and that's quite a nice base as well. But the beaches are better on the A.C. Your choice on mode of transport, but I wouldn't rent a car here either. The distracting views are almost as beautiful as Taormina, and the traffic that time of year will just start being horrendous, on and off the peninsula. Trains & buses around here are very efficient, so are ferries & hydrofoils. Best to stick to those.
Then the day before your flight out of Rome, take a train and stay in town overnight just to be safe from a timing standpoint. You never know in Italy!