There are so many lovely places to go even with out driving too far, you have all the North East and North West coasts to explore. loking further a field, I would go to Corfu Town to see the buildings, the forts and just to soak up the wonderful atmosphere. If you want a quaint village and gorgeous beach then I think Pelekas and Glyfada beach will fit the bill.
If you are prepared to drive further, there is a beautiful drive via Sinarades/Ano Garouna to Ag Matheos and the lake/beach at Korrission. Stop at Gardiki castle as you go past, there is not much to see but it is a lovely spot. The drive along the coast road from Corfu Town to Benitses is lovely too. If you have a head for heights then take the central road over Ag Deka to Messonghi or again to Korrision.
As for probably the most idyllic spot on the island, Agni bay takes some beating.
Have a look at the guides on this site, get yourselves a map and have a great time exploring.