What do you plan to do while staying in Palermo and Catania? That would govern how much time you'd need in each place. I think nearly all of us would recommend Siracusa rather than Catania for a 4-night stay, because we'd want time for both the historic Ortygia selection of Siracusa and for the Baroque towns in that corner of the island (Ragusa, Noto, Scicli, Modica).
However, Siracusa is much farther from Taormina than Catania is. Opinion on Taormina seems rather sharply divided. It is beautiful, though expensive. Many people like it a lot and would gladly stay there for multiple days. Others are like me and find a few hours on the main street (tourist-clogged and lined with international boutiques) more than enough time.
I haven't been to Agrigento because I'm just not interested in Greek temple ruins, but most visitors are different from me, and I think that detour would be worthwhile for them. I liked the Villa Romana del Casale a lot. What I do not know is how practical it would be to go to both of those places on the way from Palermo to Catania. I recommend using ViaMichelin.com to get the estimated driving time. Understand that it will not include stops, navigational errors, traffic delays, etc.
And what about Monreale? It's arguably the No. 1 sight in all of Sicily. You can get there by public bus from Palermo, by tourist bus, or you can drive there after you pick up your car and begin to head away from Palermo. But I am very doubtful about seeing Monreale as well as Agrigento and the Villa Romana on the way to Catania.
Note: You should not under any circumstances drive in the city of Palermo. It will take years off your life. See the sights in Palermo, take a day trip to Cefalu by train if that's on your itinerary, and then pick up the car for the rest of your trip.