Advise on Sicily

It depends exactly what you want to see, but personally I think Siracusa makes for a better base to a greater number of interesting sites (Noto, Ragusa, Modica, Scicli, and even Taormina). Plus, there is more to do in Siracusa itself than Taormina, which is tiny in comparison. Taormina is also much more expensive than anywhere else in Sicily (in terms of lodging, food, shopping, etc), in case that is a consideration.

I agree with Agnes. Plus I found Taormina painfully mobbed in 2015. I think you'll be unpleasantly shocked by the massive increase in visitor load since your earlier trip. I was glad to escape after a few hours, though I'm sure it's a lot more pleasant early in the morning and in the evening. But however pleasant or unpleasant you find Taormina in 2020, I think you won't get the best out of Sicily (which has easily 2 or 3 weeks of sights scattered all over) by basing in Taormina for a week. My common recommendation for people thinking of spending a week in Sicily is "Why don't you consider Puglia instead".

Trying to fill a week with viable side-trips from Taormina might be quite a challenge. You have Catania and Mt. Etna, and a Sicilian told me Acireale is nice (but I didn't have time to go there). I assume there's a bus or van day-trip or two available to nearby towns. You might see what you can find on Viator (then try to book directly with the company actually offering the tour).

Taormina has that wonderful hillside location, but that means you are a bus ride away from the train station. I think that would get annoying if you were making a lot of day-trips by train. I'd much rather day-trip to Taormina (which is what I did) than take a bunch of day-trips from Taormina.

IME a car is very helpful in Sicily. If you are short on time (and you are) and you are hoping to get very far away from your base (I suspect you will), you will get a lot more out of your time if you have a car. It's not a place where you can quickly and efficiently connect a lot of places via public transit.

One counterpoint to acraven's otherwise spot-on suggestions above, is about basing in Taormina versus day-tripping there. It's been years since I was there (and I recall it was crowded even back then), but one of the best ways to mitigate places that are too crowded is to stay there, but get outta Dodge during mid-day (since that's when the day-trippers flood the place), and enjoy the (relatively) lighter crowding back at your base in the early morning and evening. Of course, that assumes that a significant portion of the crowds are day-trippers, not sleeping there. And anyplace that gets cruise ships...those places are at their best early and late.

I would also agree with Agnes and acraven. We visited Eastern Sicily in 2014 and when our tour ended in Taormina we went to Ortigia for 4 nights and did a day trip to the lovely baroque towns of Noto, Ragusa and Modica. As mentioned it makes a great base. We are now planning to go back for several weeks next year to add areas we missed as well as revisit magical Ortigia.

Taormina was our least favorite place on our RS tour this year. Overcrowded, expensive. Ortigia was our favorite. Stay there. But as others have said a week is a very short time in Sicily. Why not skip Florence for another time?

We toured most of Sicily and our least favorite place was Taormina. We will not return there, mobbed, over touristed. It’s in a beautiful setting, such a shame.
Ortigia Island, Siracusa was our favorite place. We stayed on western side overlooking the bay and yachts from all over the world. We had almost three weeks.