Hi Beth --
I have not been to Savoca so can't help there. I have been to Acireale and cannot recommend that. I know opinions can vary, but we found it to be a pretty dead place. Not easily accessible by train -- the station is a ways outside of town, although there is a bus stop in town. Our issue with Acireale was not just a lack of things to do or see, but the "vibe" was off. We happily spent four nights in Sougia so you know we don't get bored easily.
In Taormina, we took a private taxi tour to Mt Etna. We stopped at a little bakery along the way (and got a tour and free bag of cookies from the sweet little baker), visited a honey producer, walked a little ways up Mt Etna (it was cloudy & foggy so not the best day for views but still enjoyable), stopped at a winery for lunch on the way home and then at an orchard afterward where we picked some lemons. I suspect this is a common itinerary; it was not terribly pricey. I can try to find our driver's name if you like. I think you can choose to do more of a hike up Etna than we did.
On another day my husband walked up to Castelmola. It's quite a hike so I declined, but he enjoyed it. Maybe a 45 minute hike? We also took the funicular down to the beach at Mazzaro one day and walked over to Isola Bella. The public gardens are nice ... there are operas ... the amphitheater of course ... we found plenty to keep us entertained in Taormina.