Canterbury sits at the centre of east Kent, which is small enough that you can do almost any of it as a day trip. Whitstable — fifteen minutes on the train — is the obvious first choice: a low-key coast town with oyster shacks, weatherboarded houses, and a working harbour. Dover is twenty-five minutes away and has the white cliffs, the castle, and one of the best preserved Roman lighthouses in Europe. Ramsgate and Broadstairs are forty minutes east and offer a slower, more Edwardian coast.
Inland, Leeds Castle is forty minutes by car and one of the most photogenic moated castles in England. Sissinghurst Garden is an hour west and worth the trip for the Vita Sackville-West restoration alone.
For a longer day, Sandwich — half an hour away — is an extraordinarily well-preserved medieval town that most tourists skip, with a working harbour and miles of golf-links beach beyond.