I recently returned from a 22 day stay in Sicily during Sept to Oct timeframe. This was my second trip to Sicily. On my first I only visited Palermo province, to visit the small town my grandparents immigrated from in 1913. This time with three weeks we circled the island, starting and ending in Catania and travelling counterclockwise. We used public transportation, both busses and trains, all easy, comfortable and cheap, with the exception of one day with private driver (that I saw recommended here on this forum) and who took us from Agrigento to Ortygia/Siracusa via Piazza Armerina, Villa Casale and Caltagirone.. With your four weeks you have plenty of time to circle the island and visit what most interests you. During our trip we stayed at BnBs in Catania, Taormina, Palermo, Mazara del Vallo, and Agrigento. In Cefalu at a hotel and in Oliveri and Ortygia at AirBnB apartments. I have no complaints about any of them. I always look for breakfast included when possible. I found Sicily to be very budget friendly, especially lodging and transportation. We were two women, both past 66 and we felt safe at all times. Situational awareness and common sense are important, regardless of where you are. Don't let the dire warnings of scams and pickpockets that often appear on forums scare you unnecessarily. If you are comfortable driving in Sicily, a car gives you great freedom to do more exploring but there are very good public transportation options available in Sicily. I would avoid driving in Palermo. We also used a driver one day to take us on a daytrip from Palermo to Segesta/Erice/Marsala and back to Palermo.
One of my favorite stops on this trip was the fishing town of Mazara del Vallo, with its mix of Arab, Jewish and Christian influences. We stayed two nights there and used it as our jumping off place to visit the beautiful ruins at Selinunte (which I liked actually more than Agrigento). From Mazara del Vallo we took the train to Castelvetrano and a bus from the train station down to the extensive ruins and then back to Mazara del Vallo at the end of the day.
I always try to mix both major attractions with lesser known sites and smaller towns. These are the towns we based in: Catania, Taormina, Olivera ( in order to visit Tindari), Cefalu, Palermo, Mazara del Vallo, Agrigento and Ortygia. If you have a car (or even by bus from Cefalu) head inland into the mountains, maybe to Castelbuono or Caccamo. We spent our biggest block of time ( six nights) in Palermo and were busy everyday. There are many great daytrips you can do from there and Palermo itself has many great sites and experiences to see and do. Don't skip Monreale. If I had extra time I would consider stays in Ragusa and Trapani and maybe Sciacca or somewhere inland. Make sure you attend a puppet show. That was fun.
If you want the names of the BnBs or the two drivers we used just send me a PM.