The only place where traffic is notoriously chaotic is Palermo, and Catania. Therefore avoid renting cars while you are staying there. In the rest of the island roads will have zero to little traffic.
If I were you I would fly to Palermo (PMO) and spend the first 4 nights there to visit Palermo and the Basilica of Monreale. Cefalu is an easy train day trip from the Palermo Central station. Find accommodations north of the Central station and south of the Teatro Politeama, anywhere more or less along the Via Maqueda corridor. It’s a very elegant part of Palermo and very safe.
Once you are done with Palermo, rent a car in the city and get out of town. Head west first. I stayed in Scopello a very quaint small village next to the Zingaro Reserve (great beaches, albeit pebbly and cliffs, no sand beaches). From Scopello or anywhere else in the area (Castellammare is also nice), you can take daily driving trips to Erice, Trapani, Segesta, San Vito, Selinunte, wine country (the province of Trapani produces the most grapes in Italy), even the Egadi (ferry from Trapani). Everything is within 30-60 min radius). Stay there nights that you deem necessary in western Sicily.
Then take the southern highway via Sciacca and Agrigento. Visit the Valley of the Temples in Agrigento. If you like beaches see also the Stairs of the Turks beach. You could do it all in a day without spending the night, but if you have time you could spend a night or two in Agrigento.
From Agrigento head to Piazza Armerina and visit the Roman Villa del Casale. You might want to spend one night there too.
Next head to Ortigia. Cars are not allowed in parts of the island but there is a large parking structure where you can park. Hereis the parking place. Ortigia can be used as a base to visit Noto, Ragusa, Modica, unless you want to spend nights there. If you prefer an agriturismo outside of town instead of Ortigia, I can suggest one.
Finally head to Giardini. 3 nights are more than enough. Taormina can be seen in half a day.
Drive to Catania for your last night and return the car as soon as you get there. Next day you go home from Catania’s Vincenzo Bellini International airport (CTA)