Driving in Sicily and home bases in April

We've been to Sicily 5 times, each time for a 3 weeks or so. Driving between places is absolutely not a problem because there isn't a lot of traffic. Your idea of staying outside cities and using public transport to go in is correct, Palermo in particular is a nightmare by car. The only thing to be aware of is that in even small towns in Sicily the road system in the centre usually consists of a grid/checkerboard of quite narrow one way streets going in alternate directions. This means that you have to be very careful each time you meet a crossroads (every few metres) as to who has right of way and therefore proceed slowly. So even in this case the best option is to park as soon as you arrive and then walk! PS If you stay three days in Ragusa, visit Modica .......it's beautiful!

You will be missing a lot of Sicily by spending 9 of your 14 days within a one hour drive of Palermo, just daytripping to various places and rushing back out. I think you'll enjoy the trip a lot more if you give yourself a little time to relax in a piazza and soak in the atmosphere.

You might consider staying in the historic district of Palermo for 3 nights (day tripping by train to Cefalu) and then picking up a car as you leave town. Head south to Agrigento or west to Trapani, depending on the sights you plan to see.

Ideally you could thus find a few days for Ortygia, which is the favorite place in Sicily for many travelers. Then make your way back to the Palermo airport -- or, even better, fly out of Catania.

You are missing Ortigia, a real gem. There is parking on the north end of the island as well as right before you cross the bridge in Siracusa.
As far as driving in Sicily, you need to be aware that stop signs and traffic lights are considered
“suggestions” by locals so do not assume anything. Also at traffic circles, there are different approaches.
Are you stopping at Segesta? And also Selinunte before Agrigento?

Rather than stay in Palermo, we stayed in Monreale and took the bus in. We selected hotels/B&Bs that had parking. When we had to drive in a town, I Googled public parking lots and plugged them into my GPS. Saved a lot of time wandering around looking for them. On Ortigia we parked in the lot on the north end Suki mentioned. Have €1 or €2 coins since the parking machine didn’t take smaller coin denominations on Ortigia. In Noto, there was a person collecting the fee. In Agrigento, we were able to walk to the Valley of the Temples so didn’t have to move the car.