How to spend 10 days on Sicily's Ionian Coast in late October

Sicily is on my next up list
The southeast, covered recently in the NYT, would be great in October and using public transport.
https://www.nytimes.com/2019/09/05/travel/what-to-do-36-hours-in-syracuse-sicily.html
https://www.nytimes.com/2019/05/06/t-magazine/val-di-noto-sicily-travel-guide.html
I would not count out Catania:
https://www.nytimes.com/2018/05/11/travel/catania-sicily-budget-frugal.html

Last year we spent a week on Ortigia Island off of Siracusa which is the ancient heart of the city. Fabulous.
We went to the local markets, walked through Piazza Doumo daily, ate at great restaurants and loved looking at the sea. It is an easy trip there from Taormina. We also spent a night in Noto, one of the three famous Baroque towns in easy reach of Siracusa. We drove but you can take a train or bus also.
You can book tours to Agrigento (a must see in Sicily!) easier from Taormina but possible from Siracusa as well. Driving to Agrigento, make sure a tour includes a visit to Villa Romana Casale with it's incredible mosaics.
Two nights in Taormina was enough for us. It was different from any other place we visited in Sicily, and not in a good way, very crowded and over touristed. Packed with cruise ship mobs. But it is beautiful with views of the sea and snow capped Mt. Etna. We did not visit Mt. Etna but loved looking at it every day.
We used Select Italy for transfers and they also offer tours. We had been to Palermo and Trapani previously so skipped them and spent two and a half weeks exploring in Sicily by rental car except for Siracusa/Ortigia and Taormina.. We loved this trip!

Do not Forget Palermo, Catania, Taormina, le gole dell’alcantara, the historical trains of taste, in Italian ‘ Trenì storici del gusto, la riserva dello zingaro. If you have time, Erice in the province of Trapani. You can also go and puck up chestnuts in the Madonie or Parco del Etna, Ragusa, Modica, go to the Beach,
visit Eolean islands pick up capers, eat pasta alla Norma, eat loads of figs.

Now we are thinking that we won't be experiencing good beach weather.
Even if the weather is good, depending on the beach it may still be closed. I'm not much of a vacation beach person, but when my friends wanted to go, we found that many beaches were not yet open in May and may well be closed already by October 24th.

Siracusa/Ortigia is well worth 3 or 4 nights, especially if you like to wander the old town and walk along the waterfronts. Ragusa is worthy of at least a day and made me happy as a 2 night relaxing stay with dinner in the old town.

Sicily is an island so there are plenty of open free beaches. You just need to take your towel or even buy a cheap camp bed if you prefer and take a picnic with you. It is true however that if you want to find privately owned beaches with restaurants or coffee bars open on the stop, it might be difficult after 10th November. October is still warm in Sicily, often around 27-35 degrees Celsius.

Beach concessions will likely be closed after October 1, however nobody prevents you from being on the beach in October. There will just be no concessions to rent you an umbrella and two chairs, but you can still buy your own beach towel and spend your time there, if weather permits.

Thank you so very much those of you who took the time to make travel suggestions to a Sicily "newbie!" After reading all of your ideas, I've totally thrown out our original itinerary. Several days will now be spent in Siracusa/Ortigia as well as Ragusa, Noto and Agrigento. I got my hands on a copy of Rick Steves' new Sicily handbook which contains all the information my husband and I need to navigate that beautiful island. Thanks again fellow travelers!

I often go to Taormina in October and the beach restaurants et close just after the all saints holiday in early November

I love all the great suggestions. We will be arriving in Palermo on the 25th of October and departing on the 8th of November. I also agree that Rick's book is a must have.