You can't go wrong spending a week in Rome, but if you want to go to Sicily, skip everything else and head directly to Palermo (or Catania). We've been a few times roughly circumnavigating the island, basing in a few different cities. I loved Palermo with it's history, architecture and wine and food, although the food and wine is excellent everywhere. From there, you can day trip to Monreale. We spent time in Cefalu, mostly for a relaxing few days; it could certainly be fit into one day from Palermo. Following Palermo, we stayed a few days in Petralia Soprana. The views getting to and from there are nice, but in October it will be very quiet, grey and chilly. I'd skip it in October, but the summer has real potential. Lots of wrought iron balconies, flowers on grey stone houses. On a different trip, we based part of our time in Trapani, which is not at all touristy. Small, but interesting, and a good place from which to day trip to the salt flats, Mazara del Vallo, Erice, Marsala, Segesta, Selinunte, Favignana (in the Egadi Islands), Poggioreale and Burro di Cretti.
Other favorite stops (but not in this order, on different trips) were Ragusa (be sure to squeeze in a couple hours at Cinibro Carrettieri - where they hand-make traditional Sicilian carts), Noto (day trip from Siracusa, I think), Catania and Siracusa, Villa Romana del Casale (on a drive en route to Catania; but better to arrive early or late - mid-day is a circus, and not a pleasant one), and Agrigento. We liked all the archeological sites, but of the three, Agrigento was, by far, the most crowded and touristy; lots of large groups and a cacophony of languages. However, they have a good museum and the others don't. We also visited Modica and Caltigirone, but were less impressed. Our Mt. Etna trip was a bust -- we don't hike/walk/adventure travel, so we planned to take the train, spend a few hours in Randazzo and return, but the train leaving Catania was late so we didn't have time to anything other than ride the train. It did look nice from a distance. though. I think Catania is underrated, but I love a good, gritty city, so perhaps I'm not the best judge. We were there twice an managed to miss the WWII museum twice due to poor planning on my part and an electrical failure on the other. Based on what I've read and heard, I'd go back just for that.
We're not much for beaches, but I understand that there are a couple nice ones easily reached from Palermo or Trapani. Even with several trips under our belt, we'd still return. Plenty more to see, especially in the interior, and the Aeolian Islands. With only two weeks, I think you'd have to be very picky. Maybe a week in Siracusa/Catania and the second in Palermo. You can day trip from these cities, but all have enough to keep you busy for a week just by themselves. Or, two weeks in one with more day trips. We found Lonely Planet, the RS book and https://www.visitsicily.info/en/ helpful in planning, but there's a lot more out there if you look. Your biggest problem would be the short amount of time you'd be there, but don't let that deter you. It's a great place.
