Well, what you do is really personal taste but I will give you my ideas. For Palermo, you can walk from the ship into the center of town or use public transit. Google as place called tha Capuchin Catacombs. The place is quite amazing and macabre. Its on the far side of town from the port, but you can use the local buses to get there (just ask the tourist folks when you get off the ship) or take a taxi.
As to Civitavecchia, there is a free port shuttle that takes you from ship to the port entrance. From there, its a 4 block walk (towards your right along the water) to the local train station where you can catch the train to Rome. In Rome, you are on your own and there is a ton of info on-line and in books to help you plan your day.
Ajaccio is a place where you can walk from tender pier into the small town. There is a nice sandy beach off to your right (as you walk up to town) where you can spend a nice beach day. Cafes and restaurants nearby will take care of your lunch. If you are interested in geology and seeing the island, take a tour (if offered) to the Calanches (interesting rock formations).
Alghero - you can explore the old town on foot. Its also a good place for water-based activities.
Mahon - A very interesting place that my wife and I would love to re-visit. When Franco ruled Spain, he gave lots of money to the sister island of Majorca, but hated Menorca (gave them nothing) because they were politically opposed to his administration. As a result, Majorca is overdeveloped and Menorca is still relatively unspoiled (but its now being developed). Personally, we prefer to rent a car in Mahon and drive around the island. There are some gorgeous remote beaches which can be fun (to get to one we had to literally drive across a cow pasture and close the gates behind us) and with a car you can see the entire island.
Hope this helps.