Specifically regarding public transport. Sicily is not blessed with the extensive rail system the North has, there are no high speed trains, and trains do not run as frequently. But you can get around to most any city by train or bus, which the bus network is more extensive and frequent.
Where a car becomes handy, is for parts of the South coast and interior, and due to issues and frequency, if you are trying to maximize your time. Another drawback of public transport is that it is designed for the locals, to get from town to town, so if a tourist sight is outside a town, you are usually left to a taxi, walking, or a tour that includes transport. I saw this regarding the Salt pans near Trapani, the Temples of Segesta, Villa Romana del Casale, even the Valley of the Temples, and a few other scattered sights.
But many of the towns talked about on here. Trapani, Palermo, Cefalu, Taormina, Catania, Syracuse/Ortigia, Noto, Modica, Ragusa...are all easily accessed by train and/or bus. The island is also relatively small, a 3-4 hour direct bus can get you from one end to the other, but any travel day will likely take a good chunk of the day.
Go, it will be great, just plan a slower pace and plan out the timing of your travels.
I should add, as examples, we have been to Sicily twice, the first time for just over two weeks, we rented a car for a week of that, for the central and south coast. We recently were there for just over 3 weeks, with just public transport.
