Hire an open-top car and go for a drive. The scenery is stunning, and one of the most vivid memories I have of Corfu is the scent of the lemon trees as we wound our way around the coast. There are some very secluded beaches with no more than a shack serving drinks.. very idyllic! "Paleo," as mentioned earlier, is beautiful, but can be very busy. You could try Kalami, which is very picturesque and has some historical interest.
You can also drive up to Pandokrator - the views are amazing.
If you'd rather not drive, Corfu Town itself is charming. It has, I believe, the only cricket green in Greece, as well as some charming cobbled streets and beautiful architecture. It is easy to spend some time wandering around. You should also try some of the local ginger beer - it is very fresh and has a real kick.
Mark