We've stayed, twice, near Xi and always rent a car.
Loved Fiskado and Assos. Myrtos beach. Take a walk around Argostoli. The self hire boats are fun. Take you, a picnic, swimmies and suntan cream and head off to explore the little beaches. We've not been over to Skala way much, so not sure whats there. It's a beautiful island and I love it.
On our first trip we atayed at the Cephalonia Palace and my hubby was a padi qualified diver. At the museum in Fiskado he met a group who were diving for research, so he ended up spending a lot of time out with them and the children and I spent the day in the town and nearby beaches. That was the year I bought my necklace like the one in Captain Corellis Mandolin from Garbis jewellers in Argostoli. You can tell how long ago this all was!
We returned not too many years ago when the children were teenagers and stayed at the new hotel near Xi with our own mini pool! Great place and lovely food, although my son still bangs on about the first meal we had there at the taverna opposite the Cephanlonia Palace. Its was lovely - we were all starving after the plane journey and couldn't wait for the half board evening meal!
I've not been in May - June was the earliest, so not sure, but I'd love it there in any weather to be honest.