I am on my second trip now in Sicily, both were/are about 3 weeks each.
Ortigia would work well as one base. There is plenty to do for a few days, touristy, but low key, easy for a kids and stroller. Plenty of apartment type accomodations on Booking.com.
For your other base, if beach appeals, then maybe the South coast, the area near Agrigento. There you have the temples, as well as lots of sandy beaches. We stayed at a place that was a bit out in the country, but an easy drive to the beaches and the temples. The place also had a swimming pool, (shared with one other unit) as do a number of other properties. The town of San Leone was close, with lots of restaurants, beach, beach clubs with chairs/umbrellas to rent and food service.
Enroute between bases, you might be able to visit the Baroque hill town of Ragusa (The old town Ragusa Ibla) or Modica or Noto. On the way back to Catania, the Villa Romana del Casale is well worth a stop. Without stops, the drive time between bases and/or Catania is only 2 hours max.
Palermo has some nice things to do, but honestly is a bustling, gritty, dirty city. It does have it's charm though. Cefalu is certainly a decent beach town, compact, everything right there. By car, it is maybe 2 to 2 1/2 hours from Catania, a bit further (3+) from Ortigia.
If you did Ortigia/Cefalu you would not need the car while there, except to day trip away from the town. If you went into Palermo from Cefalu, train would be your best option, you do not want to drive in Palermo, where driving is a contact sport.
Edit to add: Ortigia/Syracuse does not have beaches in town, just some small rocky coves below the seawalls. Taormina being on a hill/mountain side also does not, and the shore below is rocky. There are beaches though within an easy drive. Do check out parking and ZTLs before committing to any place.