I will be driving from Agrigento to Taormina in a few days. Is it feasible to stop in Enna, Piazza Armerina AND Caltagirone. Or is that alittle too ambitious?

Maybe Enna and Piazza Armerina, but not all three unless you are prepared for a long day of driving that may not leave you all that much time in any of the three places.

Piazza Armerina is not far off the autoroute from Enna to Catania. Caltagirone is past Piazza Armerina and the road from there to Catania is not a fast one.

We stopped in Caltagirone in the morning for an hour or two, then another two hours at the Roman villa at Piazza Armerina as soon as it opened after lunch.
Without stopping at Enna, it was a long day but very much worth it to this ceramic and mosaic fan. We did it in March, and you are at summer solstice, so you'll have longer daylight than we did. It will make for an easier trip than we had--we took a wrong autostrada exit and found ourselves in Catania as darkness fell. With help we were soon on the right road to Taormina, but I wouldn't want to repeat that portion of my trip.

So would my best bet be to head across from Agrigento to Enna, down to Piazza Armerina, back up to the A19 across to Catania... then from there drive up along the coast?

I think that would be the fastest and most convenient.

That would be too ambitious. We just got back from Sicily this weekend (it was hot, hot, hot, by the way!).
There are road works going on on the tollroad between Catania and Taormina. Because of that, and the usual traffic jams during the week, it took us nearly four hours to get from Siracusa to Taormina last Thursday (220 km or so).

Are there any alternate routes, or is the A18 it?

I think the toll road is it. But how strange! I did the same drive - a Sicilian was driving, not me! - on May 27, and there were no road works between Catania and Taormina. We did Siracusa to Taormina in 2 hours.

Eloise, they were renewing the road surface and there was only one lane that could be used for at least 2 km (which, on peak hour is LOOOONG). A bit further an accident had happened. This slows down traffic tremendously.