We were in Palermo in May 2000 for a day. We went on a half day tour of Solunto and Bagheria (spelling?). The tour of Solunto we found very interesting. There is a 30 minute or less bus ride up a mountain and at Solunto there are ruins from the Phoenician days that are being excavated there. If you like archeology you will find it fascinating. However, be aware that there is a long walk up hill, after you arrive at the site, just to get to the place where the excavation is. The view of the Mediterranean there is fantastic too. Then we went to Bagheria, a small village to view the Villa of the Monsters. Pretty disappointing in my view. It's an old villa that is not now in good shape, and it was created by some sort of mad man who carved monsters on all the roof tops and walls surrounding the villa. Interesting to see a photo of, but not to spend an hour walking through. We also stopped at a gelato shop and that was great. The Sicilians are famous for their gelato desserts.
We did that in the afternoon. In the AM we walked from the ship into town and visited a few shops where they sell pottery products. We bought a tea set there. Other than that, there is not a lot to do in town. The streets are crowded and small, and after you walk a few blocks, there's not too much else to see without taking an excursion. I recall our bus took almost an hour to get back to the ship due to congested streets. We passed an old church on the way back that had been another 'excursion' some folks took.
By the way, the weather was beautiful when we were there mid-May. Have a great time!