...and private car on a Sunday.
I am living proof that you can see Sicilia by public transportation (mostly). The better you plan, the more successful you will be. I made a couple of planning boo-boos.
Because of the Terminal 3 fire at Fiumicino, I took the train from FCO to Napoli and spent the night there. The next morning, with no air transportation possible, I took the train to Palermo, arriving 8 hours later. The second-class compartment was comfortable enough, even with all seats taken. At Palermo Centrale, I was too tired to figure out bus transportation to my B&B, so took an overpriced taxi (but the driver did lose his place in the queue, I am sympathetic). Four great nights at B&B Federico Secondo in the heart of Palermo, daytrip to Monreale by city bus, visited every place on my list except the Regional Archeological Museum, which is closed for the rest of the year (the metopes from Selinunte are apparently on loan to Expo Milano).
I left Palermo for Selinunte on Thursday. I was able to get to the station at Castelvetrano by train, but once I arrived there, the station was deserted, no taxis or buses in sight. I could not get through to my B&B in Marinella di Selinunte, and I panicked, called booking,com (who did get in touch with the B&B, they arranged for a private transfer, €30, 30 minutes). While I was waiting, the train station agent came back from a long lunch, so my panic was unfounded. I got to the B&B, and once I was ensconced in Giovanni's (and Rudy the cat) coddling, things took a turn for the better. We had coffee on the terrace with a view of the temples (scaffolded). The B&B is Terrazzo di Venere, lovely place.
I left Selinunto for Agrigento on a Sunday, there was no option other than a private car or backtracking via private transfer to Castelvetrano and who knows where from there. I opted for the private transfer ($70 door-to-door, a little over an hour) to save time and sanity. Four nights in Agrigento, great B&B Porta Atenea, at the foot of the main street. I went to the Valle dei Templi the next day, but the ticket office and entrance have been moved recently, so I walked about 300 meters to the new entrance - not conveniently signed, but if you stay on the bus, it will stop there. The entire archeological zone is now one park with a footbridge over the busy highway. You can still exit the park at the old entrance/exit, and catch the bus across the street from the bar. Great new restaurant in Agrigento, Sal8, steps from the B&B.
On Thursday I took the bus from Agrigento to Piazza Armerina, with a change in Enna. I called the B&B in Piazza to let them know I was on my way, and the owner happened to be in Enna, so he picked me up and drove me to the B&B Dimora del Conte on via Roma. Lovely, family-run place, the son and daughter prepare breakfast every morning. I visited the Villa Romana deal Casale via city bus; there is also a bus to Aidone, within walking distance to Morgantina, and with a nice museum. Filippo (father) very graciously drove me to the bus stop on Sunday morning at 7:15 (will I ever learn?), bought breakfast at the only bar open on Sunday.
I connected to the bus for Siracusa in Catania (Catania is like Purgatory, everyone goes there, but you don't have to stay long). Bus was on time, got to my B&B in Ortigia, and had another great experience. The navetta (little bus) used to be free, now it's .50€, loops all around Ortigia and to the train/bus station in the new town. Buy ticket on board from the quirky machine, only takes specific coins and does not run on Sundays.
I love Siracusa! It is my last stop in Sicilia (Catania is next, two nights and then fly to Roma). I expect smooth public transportation to Catania (buses almost every hour during the week. B&B Via Della Giudecca, manager marked gluten-free restaurant options on the map for me. Gluten-free is easy everywhere in Italy,