We have only three (3) days in Sicily. We don't want a big city, hectic, tourist traffic-choked experience (maybe like Palermo might be?). Instead, we want a more relaxed 3 days mostly in small to medium sized towns (maybe the Taormina vicinity?) moving around at our own pace, enjoying local scenery, people and food. We do want to see Mt Etna, just not on a tour bus. We may have our own car, and can use local transportation to move around. Where should we go? (I should add that our arrival airport will be dictated by the area we choose to visit. From Sicily we will drive up the Amalfi coast for 5-6 days and fly home from Naples.)
It depends where you're planning to fly into. If Catania, then base yourself in a smaller town in that area (e.g. Giardino Naxos, Taormina, etc.). If Palermo, then base yourself somewhere on that side of the island (anywhere on the outskirts of the city). Just be very realistic on what you can accomplish in only 3 days and set your geographical reach accordingly (most folks on this forum would not even recommend flying in for only 3 days, as it's incredibly short). If something is a "must" (like Mt. Etna near Catania or Monreale near Palermo, then set aside a full day for it to make sure you don't miss it). Public transit - buses and trains - work just fine (albeit a bit slow) in Sicily but not on Sundays, so keep that in mind. Depending on the departure point, it makes more sense to use a bus instead of a train. This was the case for Taormina, since the bus drops you right by the gate of the town as opposed to the bottom of the hill leading up to the town (the train). You will be at mercy of their schedules, but I've had no issues and found fairly frequent service to where I wanted to go. I visited the southeast corner plus Taormina.
I can't offer any advice on Mt. Etna since I didn't go, only that there are many different types of day tours which vary in how high you will get on the volcano. I believe you can even take public transit there, but you'll likely need some kind of guide to climb the volcano. Just do a Google search or look at Trip Advisor...there is lots of info out there about Mt. Etna.
I would pick up a book at the library to read up on what you're potentially interested in, or check out www.bestofsicily.com. I stayed in Siracusa and found it to be a great base (but only if you're not set on seeing Mt. Etna, which is too far to the north). There is enough to do there to fill all 3 of your days, along with a one hour short side trip to lovely Noto. I took a bus to Siracusa straight from the Catania Airport. It was very inexpensive, easy, and stress-free.
You really need to read a guidebook and decide whether it makes any sense for you to go to Sicily for such a painfully short period of time. If it's just Mt. Etna that intrigues you, fine, but I'm betting there will be much more. I'd use those three days on the mainland and plan a separate trip to Sicily when I'd have enough time to see more than one small part of the island.