A agree with Agnes. May might be less busy, but Taormina was overrun on a Saturday in late June 2015, and I would not have wanted to spend 3 days there unless it was to go to the beach. But the setting is wonderful and distinctive.
I used Catania as one of my bases and liked it, but it's gritty in places and doesn't have as many top sights as Palermo. I did day-trips by public bus from Catania to the Villa Romana del Casale outside Piazza Armerina (stunning mosaics--recommended) and to Enna. I do my best to get to inland towns, because they feel less touristy. For you, Catania should be skipped unless you need it for logistical reasons.
Monreale is virtually a suburb of Palermo and is a must since you're flying into Palermo, assuming the cathedral's opening hours cooperate. Take the bus from Palermo; you don't need to spend money on a tour. Palermo itself is a big, noisy city with a wealth of baroque architecture; there are many lovely churches and other religious buildings to visit. But be prepared for a fair amount of grime. Watch Rick's Sicily video to get a feel for the atmosphere. Check the closing days and early-closing days of the sights most interesting to you. I think a lot will be closed on either Sunday or Monday (your first two days).
Cefalu is an easy day-trip from Palermo. It's picturesque but I didn't feel I needed to spend the night there. It would probably be a nice place to chill out for a couple of days if you had more time, but Cefalu is quite small and--unlike Palermo--doesn't require a lengthy visit to see the sights. So I'd visit Cefalu from Palermo rather than vice versa.
I used public buses and visited Noto and Modica on a single day from Ortygia/Siracusa. Ragusa is more time-consuming, and I didn't try to combine it with another destination. I imagine that all three would be reachable from Ortygia in the spring, but I cannot guarantee it. In mid-summer there were day-tours running to some of the inland towns from both Ortygia and Taormina (perhaps more from Taormina?), but I don't know about April and early May. The tours seemed rather expensive to me.
The hill town of Erice is accessible via funicular (or bus) from Trapani, which has a pretty historic district. I liked both a lot--preferred them to Cefalu, in fact, but they are not as convenient to visit from Palermo (over 2 hours by bus vs. less than one hour by train), and you'd need to push to see both Trapani and Erice on a single day-trip from Palermo.
I think you'll need to whittle back your target list a bit. I had 18 full days in Sicily and didn't have time for Segesta or Agrigento. However, I expect you can fit more in if you depend pretty heavily on bus tours. Just try to confirm ahead of time that they will be running at a reasonable frequency while you're there. There might be some that only go once or twice a week, and you'd need to know about their schedules in advance, before booking hotels.
Food in Sicily is really, really good. Noto is known for granita (similar to sorbet), and both granita and gelato are superb throughout the island. I found the pastries outstanding, as well.