Can we do Steve's suggested all-Sicily tour, but staying in only 3-4 places?

We stayed in Palmero, Selinunte, Agrigento, Noto, Ortigia and Taormina.
You could easily skip staying overnight in Selinunte, and Noto.

Last May we had a large group in a Villa north of Catania (Aci Castello) for 5 nights. Each car went in different directions for day trips a few times, that included Mt. Etna (easyish, 2hr drive each way, a day trip is probably good enough to see it) Noto, Ragusa (I think those were done together in one day) Siracusa - not a far trip, but the group that went there also later stayed in Ortigia for a few nights, you definitely want to plan a few nights minimum here. Of course, we were less than 30 minutes from Catania so we also spent time in the city.

Suki's list (minus Noto and Selinunte) makes sense to me if you want to keep the list to four stops. It would give you decent coverage of the island, though I'd opt for somewhere other than pricey, boutique-packed Taormina. The truth is that two weeks is a pretty bare minimum for Sicily, even with a car, so you'll probably end up having to compromise on how much you can see. Around Siracusa (stay in Ortygia) alone, you have the Baroque towns of Ragusa, Modica, Noto and Scicli. And Palermo has multiple days' worth of sights, plus the basilica out at Monreale, plus Trapani/Erice, plus Selinunte, plus Cefalu. I'd probably want five nights in Palermo and four in Ortygia at a minimum.

The Villa Romana will be the biggest challenge from the standpoint of driving time from the potential bases. It would be best to hit it in transit as you travel between Taormina or Siracusa and points west.

4 bases will,do.

Palermo: you don’t need (or want) a car there. 3 nights are good. Cefalu can be visited from Palermo by train (just about one hour), in which case you may want to add one night in Palermo.
After your Palermo stay, then it’s time to pick up your rental car in Palermo. Try to pick it up in the middle of the day because at commute time the traffic gets really bad.
Scopello: i based myself in Scopello once and from there I drove to Segesta, Erice, Trapani, Zingaro Marine Reserve, Selinunte, in a couple of day trips. If you find something in that vicinity you can hit most of western Sicily from there.
Agrigento (Valley of the Temples) and Villa del Casale (Piazza Armerina) are somewhat isolated so you might spend one night around there. Your need 3 hours each.
Siracusa/Ortigia: with a base here you can visit, besides Siracusa, also Noto, Ragusa, and even Caltagirone.
Taormina: from here you can easily visit also Mt Etna. I generally stay in Giardini Naxos, just down the hill. First because I like to stay at the beach, second because Taormina is crazy expensive (including the parking)..

We did a trip five years ago and stayed in Trapani, taking easy day trips all over western Sicily. I'm back this week from ten nights in Catania. What a great location - nothing in the eastern half of Sicily was much more than a two hour drive away, and it was wonderful to come back to the same lodging, grocery stores, and restaurants each night.

Thanks, everyone. This is really helpful, and you are all kind to take the time to assist me. Much appreciated!

If you are interested in Siracusa as a base, focus on Ortigia, a fascinating and friendly island just years from the mainland - connected by two bridges.