Airport Limousine buses run three times a day from Kansai International Airport near Osaka to Marugame (4 1/2 hours, ¥5300). At other times you can take a bus to Takamatsu (3 1/2 hours, ¥5000), then take a 25-minute ride to Marugame on the JR Line (¥540).
Marugame is served from Okayama, a major stop on the bullet train, every 30 minutes or so by Shiokaze and Nanpu limited express services (about 40 minutes, ¥2200). Local services leave less frequently and may require a transfer along the way, but are only slightly slower than the limited express. With a direct local train, or by taking a Takamatsu-bound Marine Liner and changing at Sakaide to a local train towards Utazu, you can reach Marugame in about an hour at a cost of just ¥960.
From Takamatsu, local trains are plentiful, reaching Marugame in about 25 minutes (¥540).
Marugame can be used as a base for traveling to Matsuyama (2 hours via Shiokaze limited express, ¥5700) and Kochi (2 hours via Nanpu limited express, ¥4600).
The above trips can be taken at no charge with a Japan Rail Pass. Trips entirely within Shikoku are valid with the JR Shikoku Rail Pass, known as the Shikoku Free Kippu.
Highway buses services to Marugame are offered primarily by JR Shikoku Bus and Shikoku Kousoku Bus.
Buses run to JR Marugame Station from Kobe's Sannomiya Bus Terminal (4 hours, ¥3500), and from both JR Namba and Hankyu Umeda stations in Osaka (3 1/2 hours from Umeda, ¥4200). Note that most of these buses run from Marugame in the morning and to Marugame in the afternoon and evening.