The closest domestic airport to Marugame is in Takamatsu. Limousine buses operated by Kotosan Bus run directly to Marugame (1 hr 15 min, ¥1200).
The nearest international airport is Kansai International Airport in Osaka. You can reach Marugame in as little as 3 hrs by JR using a combination of shinkansen and limited express trains (¥10450 reserved seats, no charge with the Japan Rail Pass). Cheaper is to head out by bus: for example, you could take an airport limousine bus to Takamatsu station (3 hr 30 min, ¥5150, six trips daily) and change to the JR train (25 min, ¥550).
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 min, ¥2250). 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 50-60 min for only ¥980.
From Takamatsu, local and rapid trains leave regularly towards Marugame. Direct rapid services take as little as 25 min and cost ¥550.
Marugame can be used as a base for traveling to Matsuyama (2 hr via Shiokaze limited express, ¥5860) and Kochi (2 hr via Nanpu limited express, ¥4780).
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 Shikoku Kousoku Bus.
An overnight bus runs to Marugame from Tokyo (Shinjuku Expressway Bus Terminal and Hachioji) and Yokohama (11 hr 45 min from Shinjuku, starting at ¥9400). An overnight service also runs from Meitetsu Bus Center in Nagoya (8 hr, starting at ¥5600), and from Fukuoka and Kitakyushu on the island of Kyushu (8 hr from Fukuoka, starting at ¥7200).