I spent 3.5 weeks in Sicily last Spring and traveled solely by public transportation (mostly buses), so it can certainly be done.
Keep in mind that it will take longer, to get from point A to point B, as most buses make several stops along the way.
I found the Lonely Planet Sicily guidebook very helpful. It gives transportation details at the end of each chapter under the category: "Getting There & Away"
There are several bus companies that service most major cities and towns of Sicily. I found the buses to be reliable and comfortable.
Here are our late friend Zoe's trip reports which may give you some inspiration:
I actually based my trip on Zoe's trip reports and this was my route:
Flew to Milan > flew easyjet to Palermo (day trip to Cefalù, and afternoon to Monreale) > Trapani (separate day trips to Segesta, Erice & Favignana) > Agrigento ( Valley of the Temples) > Piazza Armerina (Villa Romana del Casale) > Siracusa (stayed in Ortigia, and took day trip to Noto) > Milazzo > ferry to Lipari > Acireale (day trip to Taormina > Flew easyjet back to Milan > Home.
BTW, I highly recommend all the accommodations recommended by Zoe.
Here's one more of Zoe's reports: Sicily by public transportation
How long will your trip be?
Hope this helps!
Sicilia is beautiful, have a wonderful trip!