stop between ortigia and Taormina

From Taormina, I'd go visit Mt Etna, and the wineries there. It covers a big area.

From Taormina take the Gondola down to the beach area. We did this to get away from the crowds in Taormina. Maybe the cruise ships are not there at this time but if they arrive, it is very unpleasant.
Friends enjoyed spending time in Acireale north of Catania so maybe you could have lunch there You could stop in Catania for lunch but would have luggage locked in the car which is a risk. The good beaches are south of Ortigia.
A good source for local information on Sicily is the TripAdvisor Sicily forum, especially poster Vagabonda there. She will answer your questions. I highly recommend that you do this.
And finally, we loved Sicily but I would not spend four days in Taormina, our least favorite place.

For beach, you would probably just go a few miles to the stretch between Schiso and Riposto. Acireale is a decent stop, but I do not recall much beach in that area.

The coast south of Agusta to Siracusa looks nice in theory on a map...but the petrochemical industry has taken away the charm.

It is such a short drive that I would just go straight to Taormina. The coast south of Catania is blighted by industry, Catania itself does not lend itself to a quick stop (it's too large), and north of Catania, Taormina is the main draw.
Although Aci Castello is quite picturesque as well, it is a bit confusing to reach from the highway and probably not worth the hassle.
Also, when stopping, make sure that NOTHING is visible in the car. And if you need to put stuff away, stop somewhere else to do so before you intend to park.

As balso stated, it is a short drive. The drive from Ortigia to Taormina should take about 80 minutes at most. A quick trip to Castelmola will occupy a day. To me Taormina is worth 2 nights at most. You could tour Mt. Etna. If you like castles, there are a few in that area.

Aci Castello to Acireale and the Coast of the Cyclops makes for a scenic photogenic stop. We were there in September two years ago. We stopped in Aci Castello for a delicious granita break and a walk along the shore and then went to the center of Acireale to see that town. Lots of folks were at the beach, sunning and swimming, but it is very rocky.

Thank you to all for your suggestions