You can't see all of Sicily from just two bases, no matter how long you have. While you can see a lot of things in 10 days, that wouldn't be enough to see the entire island even if you were willing to switch hotels every night. I figure Sicily needs at least two full weeks with a rental car for part of the time or 18+ days with no car. The island is large and a very rich destination for travelers.
Since you can't see everything, it's important to create an itinerary that fits your personal preferences.
Sicily has three coasts, as I see it: north (Palermo-Cefalu), east (Taormina-Etna-Catania-Siracusa) and southwest (Agrigento). If I had to stick to two bases in Sicily, they would be Palermo and Siracusa, and it wouldn't be a difficult decision. Palermo has a lot of sights and positions you well for side trips to Monreale and Cefalu--both easy by public transportation. Ortygia, the medieval district of Siracusa, has a lot of charm, and offers interesting side trips to a number of Baroque towns in that corner of the island. The ones I've been to are Ragusa, Noto, Modica and Scicli; they can all be reached from Siracusa by bus and/or train, but note that there seem to be no inter-city buses running in Sicily on Sundays.
My choices are based on my personal interests, which do not include rural hiking or Greek temples.
Sicily lacks express trains, so you won't move rapidly from place to place, but you can reach all the places I mentioned--and a great many more--without a car. There are trains running along the north and east coast. There's not a lot of track cutting through the middle of the island. www.trenitalia.com is where to go for schedules and fares. The fares, at least, will cause you no pain.
Rick has a guide to Sicily that wasn't available when I took my trip in 2015. I urge you to get a copy of it and dig in. You're going to have to make severe compromises due to your limited time and preference for only two bases--though you will not run out of things to do by staying in Palermo and Siracusa. The guidebook will help you make the right choices for you. They may be different from mine.
