We are also traveling to Sicily in April. Our plan is to work our way around the island. We are flying in and out of Catania because that's the airport our airline (Delta) uses. We could have done Palermo as well. I heard one downside of Catania is the low probability that the volcano will create visibility issues.
To plan our trip I calculated transportation times between some of the places we are traveling to. Unless your trip is short, I think you would want multiple "bases". If you have just a week, maybe you do want to base in Palermo and thus flying in and out of Palermo makes sense. From Palermo, you have possible day trips to Trapani, Cefalu, Segesta, Erice. In my research, Catonia has far fewer things to see than Palermo. If you only base in Palermo, the villa romana, Ragusa and Siracusa are really quite far away.