Takamatsu is the main entry point for Shikoku and from here the entire island is your oyster.
Kotohira, site of the famed Kompira Shrine, is an hour away by Kotoden train.
Marugame, the site of Marugame Castle, one of Japan's few remaining original castles.
Mount Ishizuchi, the tallest peak in Shikoku, is about two hours away.
Naoshima, a small island on Seto Inland Sea with superb contemporary art museums, is one hour away by boat.
Tokushima, capital of Tokushima Prefecture, is most famous for its Awa Odori Festival.
Kochi, is famous for the beautiful Kochi Castle, which is one of Japan's few remaining original castles.
Easy day trips to the mainland:
Okayama, home to the beautiful Korakuen Garden, one of Japan's top three gardens, as well as the beginning of the Kibiji District Trail
Kurashiki, famous for the Bikan Historic District, one of Japan's largest and best preserved historic neighborhoods.
Tsuyama A wonderful little town famous for Kakuzan Park, which contains the ruins of Tsuyama Castle and thousands of cherry blossoms within its precints, the historic Joto Street, and the beautiful Shurakuen Garden.
Also, depending on timing and visa, Shanghai is an option, with Spring Airlines and its typically low fares.