Six nights is painfully short for Sicily; the island really needs more than 2 weeks (longer if you don't have a rental car). I highly recommend that you focus on a limited area of the island so you don't waste a lot of time in transit.
Cefalu is a cute former fishing village turned beach resort. Probably not terribly touristy in April, but also possibly a bit dull in terms of atmosphere--I just don't know. Taormina is mind-bogglingly touristy but drop-dead gorgeous. I'd choose Taormina over Cefalu, and I wouldn't choose to combine the two on such a short trip; in my view Sicily has more important things to see than Cefalu. I was ready to leave Taormina after a few hours because of the wall-to-wall tourists walking down the main street full of tourist shops, but it's true that many folks love it; with luck you'll be like them.
I haven't been to the Aeolian Islands but will visit them on my next trip to Sicily. I believe they are interesting, but you will burn considerable time getting there, and it's definitely not something I'd do with only 6 nights available in Sicily.
Instead, I'd go to Siracusa and stay in the medieval district of Ortygia. Ortygia's a lovely area. Siracusa itself has a nice archaeological museum and an archaeological park. And there are a bunch of Baroque villages in the interior that can be visited from Siracusa. There is some public transportation, though it would be more efficient to travel by car. I thought Ragusa was the most interesting of the towns, but Noto, Modica and Scicli are also worthwhile.
I liked Catania, and it's easy to reach from Taormina. You can either take a bus from Taormina, or you can get the shuttle bus down to the Giardini-Naxos train station and take the train. This isn't something you need to plan ahead of time, so you can wait and see how enamored your are with Taormina. Catania has a lively food market (consult your guidebook for days of operation), a nice cathedral, and an attractive historic district. There's also a WW II museum that I unfortunately missed. Catania's a large place, and much of it is nondescript.
A local recommended the coastal town of Acireale to me. It's on the rail line and not far from Taormina. Online photos look promising, but I didn't have time to go there.