Taormina is fabulous for a few days, the theatre, Isola Bella, eat at La Namauchie for gorgeous food that isn't the constant pizza or pasta that is everywhere. It was a Saint's day when we arrived and there was a marching band and all sorts, was fab.
I was a bit underwhelmed by Etna, we went to 'the top' but it's not really the top it's just the side of the top, there's no caldera just views to the sea, then we stopped a little lower which was more interesting to walk around with a smaller caldera. If you don't go to the top though then the place at the bottom where you get dropped on the tour is tacky and less inspiring. The weather was not brilliant which didn't help and I've been to Vesuvius before so maybe it didn't have the novelty factor but I wasn't blown away.
We got public transport everywhere and only passed through Catania but the area around the bus/train station was a dump and poor and we were glad not to be staying. Got the train down from Taormina, but then the bus across from Catania to Agrigento. It was easy - 4 hours maybe? Agrigento was amazing, we even liked the little town there (nothing hugely exciting but a nice base). It would be a long day trip though, I'd save it for if you go to Palermo and do a side trip from there as the train is easy to get up and down and I'd recommend staying the night there and getting to the ruins early.
We loved Palermo, stayed in an airbnb between the train station and the historic centre, many of the roads were closed off to traffic (not sure if that's a permanent/summer/Saint's day thing) and it was a great location, a very diverse local area, not wealthy but nice people and we loved it. We were there for the annual Saint Rosalia celebration and were out till midnight with crowds and crowds of people. Felt very safe, lots of families and had such an amazing evening (particularly as we had no idea it was going to be on). DH often says he could very happily go back for a city break as he really enjoyed wandering around there.