Sicily

Really, you cannot see Sicily from just two bases. It's too large, and the top sights are too widely scattered.

Sicily is not an expensive destination, and I think you'll find shorter-term apartment rentals to be quite reasonable in price (though I admit that I stayed in small hotels and B&Bs).

If had a month in Sicily, I'd be looking for a base in each of these areas:

Palermo
Trapani/Erice
Island center (Piazza Armerina, Enna, etc.)
Siracusa
Catania/Taormina

Even that doesn't position you very well for visiting the Aeolian Islands or Agrigento, though the latter could be reached from Enna if you don't mind driving for over 3 hours that day.

We're not planning on two bases. We'll be moving around for the first week or so with 2 or 3 days in each place. We'd then like to have a base in the south for day trips or an overniter or two.

I have stayed in Barcellona, a lovely town with fantastic pizza! You can reach the Aeolian islands from this area. Messina is another city To think about. Taormina is touristy but has great views and more.

acraven's point is valid. You're spending about 30 days, the first 7-10 hopping around. That leaves 3 weeks at least in one place. I'd spend 3N in Palermo (there's enough for 2 days), then 2N in Cefalu. Not sure where else you want to go "north".

Agrigento is sorta in the middle on the southern coast. It's at least a couple hours' drive to much of anywhere else. That's a really long day trip. Siracusa might be okay to explore the eastern end of the island, but you'll still have at least an hour's driving each way to get anywhere else.

I think as far as bases go Siracusa is a good choice. It’s a great town itself with Greek ruins and nice places to walk around (stay on the island of Ortygia). It’s also not far from the baroque towns like Noto, there’s plenty of beaches nearby, Catania is an hour, and Etna is 2 hours.

I’d do 3-4 nights in Palermo, a few nights somewhere west of there where you can visit Erice, Trapani, Segesta and the Coast like Scopello. Then head south for Agrigento area, then Siracusa as outlined above.

We toured Sicily last summer and loved it. We flew into Catania, Stayed in Taormina - we Loved it and Isola Bella Beach, yes they are touristy for a reason. Mt Etna, ect, We stayed 3 nights, could have added another with a hike through the lava caves. Then we headed to Cefalu for the day. Jumping off the sea wall and visiting the Lavatoio Medievale where the ladies washed the clothing were the two highlights, but I could have skipped it. We visited Segesta, very nice. It is relative two where else you have been in the world. A friend who goes to Sicily every year, recommended the rest, and we are glad he did! Trapani, a great working and tourist town, wonderful evening passeggiata's. We took a boat over to Favignana and rented motorized bikes (my sons favorite trip) and swam the amazing beaches. Also went to the Salt Flats of Saline Della Laguna. It was great for my tweens and loved it. We had booked the harvesting the fleur, but it is weather permitting where as the wellness visit was not. I would book the wellness trip, they are very accommodating. We drove to Erice (funicular was closed to wind). We ranked our other Trapani stops higher, but understand why it is popular. Then onto Castellammare del Golfo. Loved this working town, amazing dinners, amazing beaches. Flew out of Palermo. Have fun, you can't go wrong in Sicily!