In terms of time in each place, so much depends on what you want to see and whether you plan any day-trips.
I think with a car you might be able to cover Ragusa, Noto, Modica and Scicli in two day-trips from Siracusa or from Ragusa (which has more to see than the other three towns). Siracusa itself needs at least 1-1/2 days, I think. You have the archaeological museum and the archaeological park, which are near each other but not in Ortygia. Then there's Ortygia. So for me, 4 nights in Siracusa would be a minimum if you don't want a separate base in Ragusa.
Palermo has many glorious churches and chapels--enough to keep you busy for many days. But lots of folks only want to visit a very few. There are also two interesting food markets and some museums. So Palermo could be one full but massively-rushed day (2 nights) or you could really pound the pavement like I did and stay there six nights (including a day-trip to Cefalu and a partial-day-trip out to Monreale). Cefalu is touristy in season, but you're going early enough that it may not be swamped. A few hours was long enough for me; the historic district is nice if you can ignore the rows of shops selling beach toys, which I really could not. I think Rick said at one point that Cefalu was his favorite place in Sicily; we clearly have different taste.
Catania has an interesting food market and quite a pretty historic area near the cathedral and university. A day there would probably be OK for most people, and you could do that from a base in Taormina if you prefer. You can visit Mt. Etna from Catania. Depending on whether you want to go all the way to the top, I think it could take a good chunk of a day. I just went around the base of the volcano, and I used public transportation, so I don't know how the logistics would work with your own car. You can see Mt. Etna from a distance without actually taking the time to go there, which is what I would tend to recommend if you don't plan to climb to the top. I'm sure others would differ.
Taormina is beautiful but massively touristy and has relatively high hotel rates. So you need to consider your tolerance for sharing space with hordes of other tourists before deciding to make it your base for that area. A few hours was enough for me, but some folks enjoy spending multiple nights there. You can visit Mt. Etna while staying in Taormina; same comments as above. Do check on the parking situation if you choose to stay in Taormina.
It takes some time to get from Trapani to Erice, even though they are quite close together. The cable car is the picturesque way to do the ascent, though you can also drive. The base of the cable car is not near the historic port area of Trapani, so you must either drive over there and find a place to leave your car (if taking the cable car) or you must take a bus or taxi across town. I enjoyed several hours of wandering around Erice and would have enjoyed more time. It would be a nice place to spend a night, I think, because the little town gets a lot of visitors during day-tripping hours. However, the luggage challenges would be significant, so I would recommend staying in Trapani. Two nights would do it for Trapani + Erice if you didn't both arrive late and depart early. There are some other things to see in this general area--salt pans, Segesta, etc. Including them would require more time.
I spent a day in the Egadi Islands, ferrying in from Trapani. It was a pleasant-enough but underwhelming experience. I don't recommend spending the necessary time to do that. I suspect the Aeolian Islands in the northeast are considerably more worthwhile.
I do not do beaches, so my suggested timing is not intended to cover beach visits.
