I’m not a fan of Catania and would not stay there at all, unless you need to stay at the airport for a very early flight (although the drive from either Taormina or Siracusa area to the airport is not that long).
If you want to visit Mt. Etna, the excursion up there will take the best part of a day. You probably want a day to relax in Taormina and Giardini Naxos as well. So I’d stay 3 nights in Taormina if you plan a day excursion to Mt. Etna, otherwise a couple of nights are enough for Taormina.
With a car you might consider staying in Giardini Naxos, down the hill from Taormina on the beach. Find a hotel with parking. The drive up to Taormina takes minutes. Few hotels in Taormina have parking, but there is a multistory parking structure inside the mountain.
Siracusa and Noto are a-bit too far from Taormina for day tripping from there.
I’d stay there another 3 nights. With one day max devoted to Noto, and the rest to Siracusa.
If you stay only 2 nights in Taormina, maybe you can add one night here in Siracusa.
You could even stay in between the two at this Agriturismo. I enjoyed it. It’s just minutes by car from either Noto or Siracusa.
http://www.masseriasulmare.com
I think we ran out of your 6 nights. Buon viaggio.