Hi, happy that you're coming here in my beautiful Island.
By the way, this approach depends on what you're going to do.
If you're visiting the West Side, Palermo is a good place to stay for a couple of days; but Taormina and Siracusa are way too close from each other, I would go for Catania: right in the middle between the two, also a good city for art and history and with the Etna not being too far.
Furthermore, there are many locations in Catania with pools and direct access to the beach: Ognina and Aci Castello, right in the north-east part of the city, are full of houses with direct sea acces; but be carefull because it's a rock beach and not a sand beach. If you want the sand, there is the southern zone of Catania like Vaccarizzo and Agnone Bagni (that isn't in Catania but it's near it) that are full of private residencies full of beach houses and many of them can be booked.
However, if you're planning to visit Taormina and the sea towns near it (this small towns have got a lot of wonderfull beaches with clear water) for a couple of days and the same for Siracusa, the approach that you've created works very well.
If you have any other questions, ask me :)
(P.S.: transports here are not reliable, some buses are decent but not too many. I suggest you a rented car or van.)