Eight nights is kind of tough, but you can do it ok. If you want to include Ortigia, I’d fly into Catania and out of Palermo. Number of nights I’d think 3 Ortigia - 2 Cefalu - 3 Palermo. I’d really only include Cefalu nights if you want a beach day, otherwise it’s an easy enough day trip from Palermo.
There’s a case to be made for sticking to one side of Sicily with eight nights, sticking with Palermo/Cefalu/Trapani or Ortigia/Catania/Taormina.
Sicily is safe. Palermo is safe. Staying in Palermo city center is reasonable and Palermo is freaking rad. Don’t skip it. June is out of season for opera, though you may find an odd performance. I caught one in Ortigia’s fine opera house. Lovely building but subpar performance.
Bus is the primary people mover in Sicily, and you should choose it over train just about everywhere. The train system here just doesn’t compare to other Western Europe systems: Rough terrain, lack of investment, lack of demand. There’s a few good lines (Palermo to the airport and to Cefalu) but others (Ortigia to Noto) can be positively stone-age. If you can arrange it, a tour group transfer or private driver might get you across the island and include a visit to Villa Romana or Valler of the Temples. This would be a worthwhile splurge if you can find something like that. Renting a car for a day or two really changes the equation and is worth considering.
