Corfu & Archaeological Sites to visit...
Butrint - via Saranda, Albania
First option I would highlight are the Greek and Roman remains in Butrint, which is a UNESCO world heritage site and set in a recently created National Park. These are actually located in Albania and can be visited via the Albanian Port of Saranda, which is about 1 hour from Corfu Town by Ferry. Probably easiest to take a excursion package for around €60, this will include help with the entry visa and a guided tour of the site, lunch and time in Saranda before returning to Corfu. The trip can also be taken without the visit onto Butrint, so a good chance to take the rest of the family for the day out, without the history bit for those who are not interested. Also a chance to briefly visit another county and the Albanian Riviera!
Oracle at Dodona - Epirus (Mainland Greece)
Generally considered the second most important Oracle site in Greece after Delphi, plenty of remains here to visit, set in a remote and picturesque valley location. Still accessible from Corfu Town on a day trip, the ferry takes just over an hour to Igoumenista on the Greek mainland and the site is just over an hours drive from there. Probably best suited to an independent trip (car hire rules permitting).
Brief YouTube video of the site is can be seen by clicking here.
On the Island: Corfu Town & Achillion Palace
Firstly some interesting sites in and around Corfu Town itself. A joint sightseeing pass is available for around €8. This will admit you to the Old Fortress, Antivouniotissa Museum (containing Byzantine artworks) and The Archaeological Museum.
Just to the South of the town is Mon Repos Estate, a wooded parkland estate with Neoclassical Villa, it also contains the Museum of Palaeopolis (3€), another archaeological museum. Further on in the park are also the ruins of two Doric temples. It sounds like a good spot of a picnic and a wander, with a bit of history for those interested.
Finally should probably also just mention the Achillion Palace (see image), about 10km South of Corfu Town, €3. A largely 19th century creation used by Austrian Empress and later owned by Kaiser Wilhelm II. Probably more of like a Hollywood set than a genuine historical site, but might well be worth a visit.