Tell people you're heading to Matsuyama, and Dogo Onsen will be the first place they recommend. These hot springs have been the city's top attraction for the last millennium. Japan's oldest written text mentions it as a favorite of gods, emperors, and peasants alike, and it's still in daily use by locals and visitors. The main wooden building at present-day Dogo dates from 1894 and looks like a fairy-tale castle; the only thing that's changed significantly is the view.
Admission is ¥410, but for the real Dogo Onsen experience spend ¥840 for a bath with some frills. Head upstairs and you'll get a basket with towels and a lightweight yukata robe. The staff will point you to Kami-no-Yu, or the Water of the Gods. The gods apparently liked things simple: the great granite tub is plainer than modern multibath complexes, but the water feels terrific. If you want to go all out, ¥1,550 gets you into Tama-no-Yu, or Water of the Spirits, which has the same water but is usually quieter.
All baths are gender segregated. Remember proper onsen etiquette: wash and rinse yourself (and your towel) before getting into the bath.
After you bathe, relax upstairs. The ¥840 ticket includes green tea and sembei crackers, served in a serene public tatami room. The ¥1,550 option buys you botchan-dango sweets in a private room, plus a quick look at the private bath reserved for the Emperor when he visits. Kicking back in the second-story tatami area is one of the great joys of coming to Dogo. Sip free tea or explore the upstairs quarters where writer Soseki Natsume stayed and worked during his time in Matsuyama.