Planning a 10-Day Summer Holiday in Sicily –Looking for Advice

Taormina is great - pretty town with Roman arena perched high above the beach with a funicular.
From there, you can do trips to see the archaeological sites, if the mood takes you

From Palermo you can day trip to Cefalu which has beaches. Also a cute town.
We also liked Taormina, but we didn’t go to the beach there.
From Messina you could ferry to either Lipari or Salina for a night or two. Salena was very inexpensive, but again we didn’t go to the beach.
Do a little more research and I’m sure you’ll find what you are looking for, but, with 10 days I would split my time.

We found Taormina to be terribly expensive and very crowded. It is pretty, but a couple hours/half day was enough for us. On the east side, we LOVED Siracusa/Ortigia.

We did enjoy the time we spent in Randazzo near Mt. Etna. Some day trip to Etna from Catania (as well as Taormina) There is an excellent WWII museum in Catania.

From what you said, I think, for you, a split between Palermo and Siracusa would serve you well. A "beachy" day trip from Palermo would be Cefalu. It is a charming town with a long beach and a nice hike up castle ruins with stunning views.

It will be quite hot.

Depending on where you go, you may not need a car. We wouldn't have wanted to drive or park in Palermo.

I loved Taormina and did take the gondola ride down to the beach one of my days there. Town was charming. Loved the view of Mt Etna from my hotel terrace. Shopping was expensive, but fun.. However, this was over 10 years and I suspect the small town might not handle large crowds very well. I did not like Palermo at all. Graffiti was terrible and the first time I ventured out of the hotel there was a dead cat lying in the street. That probably effected my opinion. I did like the painted carts which I saw there and there were several day trips to take in the area. I also liked the beach in Cefalu. Do you have Rick's Sicily Guide? If not, you might try your local library to see what guides they carry.

If you have not made your airfare reservations yet, If I were you, I would split your trip to stay one half near Palermo and the other half near Catania, flying in/out of one or the other, or you will be spending a lot of driving. Although Taormina (closer to Catania airport) is pleasant, and I do recommend visiting, it has become overhyped, touristy, and expensive, while the rest of Sicily is much more reasonably priced. As part of our RS tour, we stayed a couple of days in charming Ortygia, and absolutely loved it.

Have you seen the "Leopard" which was a series on Netflix. It took place in the time of Garibaldi's peoples revolution to make a united Italian state. The title refers to a local Prince of the old Sicilian Nobility. His nephew supports the revolutionaries which causes conflict in the family. There is Romance, Fabulous Costumes and Great Scenery. One of the ongoing themes was the heat of the Summer. The Prince and his family always retreated to their Mansion and Estate in a mountain Village to get relief from the extreme heat. Of course, they didn't have AC back then. Most of the places were real and maybe open to view. Be sure to get a car with AC!

summer in Sicily will be pretty sweltering -- we visited twice in October and some of the days were quite warm even then..........there's a reason most tour groups don't go to Sicily in the summer.
there are some very pretty seaside areas around Marina di Ragusa, and that area isn't very far from Syracuse/Ortigia so you could easily have a lovely vacation in the eastern side of the island. along those lines, the Vendicari Reserve also has nice beach areas, and the seaside town of Marzamemi is beyond charming. the Planeta Winery resort near Menfi, La Foresteria, is a lovely property and they have a private beach with shuttle service. in October the beach was closed but we took some nice walks. the added advantage here is being close to Selinunte, which is a fascinating site of temple ruins and also an excavated city area.
closer to Palermo, we stayed a couple of nights in Castellamare del Golfo, a nice beachside town. we did a day hike in the Zingaro Reserve -- I know you mentioned not caring about that kind of thing HOWEVER, not too far along the trail there are spots where you can walk down to small pebble-beach coves and take a swim.
finally, we did not care for Cefalu beyond a quick overnight, but the beach and boardwalk might be appealing to you. they were both quite crowded in October, so I would imagine the sand would be wall-to-wall chairs and towels in the summertime.