Sicily by Car-Which Way?

Our Sicily tours have done it both ways, in different years, with the current route starting westbound. That probably has more to do with allowing people to spend after-tour time near Cefalu or Catania than with any other practical consideration.

Which direction you travel from Palermo depends on what you want to see. There are "highlights" in all directions - South (Agrigento, etc), West (Trapani, Erice, etc), and East (Cefalu, Taormina, etc). Only you can determine which of these (among many other places on the island) are most important for you. And I wouldn't restrict myself to the Autostrade just to save time. Nothing on the island is that far from anything else, particular if you plan to go in a modified loop (for example, starting in Palermo and ending at Catania, for their airports).

On my trip earlier in 2014, I saw Palermo, Monreale, rural areas near Enna, Agrigento, Villa Romana de Casale, Syracusa, Noto, Ragusa, and Taormina. My VERY detailed trip report is here: https://community.ricksteves.com/travel-forum/trip-reports/sicily-palermo-caltanisetta-siracusa-taormina

I don't think it matters. Here's what I did in late April to early May:

Day 1 Arrive Palermo late afternoon
Day 2 Palermo, Monreale by bus
Day 3 Palermo, evening train to Cefalu (left my luggage in Palermo and took a small bag for 2 nights)
Day 4 Cefalu
Day 5 Morning train to Palermo to pick up rental car, stay near Segesta 3 nights, visit archaeological sites and coastal towns
Day 8 Agrigento overnight in Piazza Armerina
Day 9 Villa Romano del Casale, drive to Syracuse (Ortigia) for 6 nights
Day 15 drop car at Catania airport and leave Sicily

With hindsight, this was a good plan. A car would have been a disadvantage in Palermo and Cefalu. I loved staying in Ortigia and Syracuse is a good base for pretty much all of the eastern side of the island. Since I was continuing my trip in Italy, I chanced flying out of Catania. Though it doesn't happen frequently, it is not unusual for Mt. Etna to spew ash and smoke which can shut down the Catania airport for hours or even a day or two. If I were planning an international flight, I'd probably head back to Palermo instead. I took my time, lingered in lots of places, had lazy mornings, and a few long dinners. I wouldn't worry about saving time. The driving distances aren't that great and the roads are good. The benefit of staying close the the coast would be stopping at beaches along the way. Since the water was still icy cold when I was there, the beaches didn't tempt me, but the water looked really inviting.

If you plan a full "circumnavigation" of the island it doesn't matter if you do it clockwise or counterclockwise. As mentioned above sights are in all directions.
From Palermo counterclockwise:
West Sicily: Zingaro Reserve, Scopello, Erice, Segesta, San Vito Lo Capo, Marsala, Selinunte, etc.
South Sicily and Interior South: Agrigento, Sciacca. Ragusa, Piazza Armerina, Caltagirone.
East Sicily: Noto, Siracusa, Taormina, Mt. Etna, Catania,
North Sicily: Cefalu, Aeolian islands.
Two weeks are adequate to see most of the above.

Two years ago we drove around Sicily for 4 weeks starting in the west from Palermo. We got our car at the Palermo airport and stayed in Trapani, Sciacca, Agrigento, Enna (could have made this a day trip rather than overnight), Piazza Armerina, Ragusa Ibla, Siracusa, Catania, Taormina, Cefalu, dropped the car at the airport and stayed in Palermo. We stayed 2-4 nights in most towns with one-nighters in Enna and Piazza Armerina. In two weeks you won't stay in each of these towns but our route might give you an idea of one way to circumnavigate Sicily. If you go to Sciacca be sure to eat the Cassata Siciliana. It's the best dessert ever. After we left Sciacca we tried this dessert in two other towns and each was a big disappointment. Other than that, we loved everything in Sicily! Hope you have a great trip.