Sicily

Siracusa without a doubt. You'll be to get to the Baroque towns of Ragusa, Noto, Modica and Scicli by train and bus. I did it in 2015. One day for Ragusa alone. Check current schedules to see how to fit the others together. It may take 1-1/2 days; Noto is small and close to Siracusa.

You can also take the train north to Catania (markets, Duomo, WWII museum). Taormina is also possible (train to Giardini-Naxos then bus up the hill), but that will put you there during the middle of the day when it is most overrun by day-trippers (think shoulder-to-shoulder mobs traipsing along the main street). Between Catania and Taormina is the coastal town of Acireale, which I think sees few foreign tourists. It was recommended to me by a Sicilian, but I didn't have time to go. Photos look promising.

However, there is much to enjoy in the historic district of Siracusa (Ortygia, which is where you should stay). There's a good archaeological museum and an archaeological park. Be sure to allow enough time to enjoy Siracusa itself.

From Palermo you should definitely go to Monreale (by bus). Cefalu is easy by train. There are some beach towns and a nature reserve to the west of Palermo. I have never been there and don't know what sort of public transportation exists.

But I wish you could bring yourself to change hotels more often so you could see more. Sicily is just too large to see from two bases, and even more so if you don't have a rental car. You can't see the whole island in two weeks, but with one additional change you could include Trapani/Erice and the Greek ruins at Segesta. Alternatively, you could make a stop in the interior (on the way from Palermo to Siracusa) at either Caltanissetta or Enna and go to Piazza Armerina to see the Villa Romana del Casale.

Warning: There are very, very few buses running from town to town on Sundays and holidays. That should be taken into consideration as you plot your itinerary. Sunday's a day to stay where you are, though I assume the local buses probably do run from Palermo to Monreale that day. There will be some train service.

YES!!!!! Siracusa. I suggest the island of Ortigia. Buses and trains to the baroque cities are not the most convenient but they are doable. The Roman and Greek ruins in Siracusa are well worthwhile - especially the Archimedes museum (and I don't like museums). This will be my 3rd year in a row going to Ortigia - for a week in June each time. It really is more interesting than I thought it would be. Amazing views, great food, very nice people, relaxing/enchanting Piazza Duomo, interesting local art and great Nero d'Avola wine. The city of Siracusa is worn down and not very touristy but the walk along the shore is beautiful. I've rented an apartment on the Piazza Duomo with a killer view of sunset. Highly recommend you find a similar view from hotel or apartment!

3rd vote for Siracusa. Stay on Ortygia, it’s just beautiful and a great place to wander. Then in the mainland Siracusa there’s the archeological site and museums, very easily reached by city bus. Further afield you can go to Etna (at ~2 hours away it’s a long day but perfectly doable), or the baroque town of Noto and others, or to Catania for big city things, or to beaches along the south coast...

I had a car but other people I talked to there did it by bus so I don’t think that would be an issue. You could also take organized day tours if that’s your thing.

For this plan it’s best to fly into Palermo and out of Catania (1 hour from Siracusa), or vice versa.

Be open to spending a night inAgrigento to
visit Sicily’s top archeological site, The Valley of the Temples.

I agree about Ortygia. If you could bus or train up the east coast to a town near Taormina, like Giardini Naxos, and spend a night or two, you can bus up to Taormina and up higher from there to Castelmola for wonderful views. We just came back from a two-week mostly driving trip (except Palermo), and stayed in a sweet B&B in Giardini Naxos for under $75 a night. From there, there is a fast bus that takes you to the airport in Catania. Would be a shame to miss Mt. Etna.