Sicily can be a little tricky relying on public transportation. We went around the entire island. If you start at Catania, you can do the train to Randazzo by Mt. Etna and then continue counterclockwise via train to Palermo. From Randazzo you would have to take the train to Riposto and then switch trains. From there on it would be pretty straightforward. Possible stops could be Taormina, Messina, Cefalu. We stayed 2 nights in Cefalu and loved it there. We wished we had additional days, we could have done some nice day trips from there.
If you want to go clockwise from Catania, you can take the train to Siracusa, Noto then Ragusa. From Ragusa on, you would have to use buses or a private driver to Palermo. Possible stops between Ragusa and Palermo would be Agrigento, Selinunte,Segesta, Mazara and Trapani. There are many possible daytrips from both Trapani and Palermo.
I found the Rick Steves guide especially helpful for planning our trip to Sicily. You could see which cities interest you and then each chapter gives you options for transportation to travel between the cities.
With 12 days using solely public transportation, you will have to plan carefully because in some areas it does take longer to get around with public transportation, especially along the southern coast.
We had a private driver take us to wineries and also to guide us around Randazzo. If that is of interest, I could provide his contact info.
