Most pilgrims only make it to the shrine at the foot of Ishizuchi, but if pulling yourself up the side of a mountain on iron chains sounds like a good time, the climb to the summit is for you.
Ishizuchi is believed to have been an active volcano about 15 million years ago. Designated as a national park in 1955, it has become a reserve for soaring falcons and sneaky dormice alike. The official climbing season is from July 1st until October; note that the first day of the season is reserved for men only! It is, however, entirely possible to climb the mountain as early as May or as late as November, although facilities en route will be limited.
In the autumn (around mid-October), the hike up the mountain is a great way to view fall colors. However, on weekends during this time, the trail towards the summit becomes crowded and a bit congested with hikers (some with dogs in tow). In the winter, a ski and snowboarding resort operates on the same grounds. The Joju shrine is open around the year and is a popular place to visit on New Year's Day.
The closest train station is Iyo-Saijō on the JR Yosan Line from Matsuyama to Takamatsu. From Okayama, the Shiokaze Limited Express train makes hourly runs to Iyo-Saijō station (1 hour 50 minutes; ¥4940). The ride from Takamatsu is 90 minutes using the Ishizuchi limited express (¥4520).
From Iyo-Saijō, there are connecting buses to the start of the ropeway (1 hour, ¥970), but there are only 4 buses daily. They're run by the Seto Inland transportation service  (tel +081 0898-23-3450). The 10:23AM bus is the last that will get you to Ishizuchi in time to make the climb and catch the last bus back (5:22PM). A detailed schedule can be found online  (in Japanese).
The Ishizuchi #7, departing Takamatsu at 8:44AM, and the Shiokaze #3, departing Okayama at 8:30AM, will allow you to meet the 10:23AM bus. From Matsuyama, the 9:12AM Shiokaze #12 will also allow you to catch this bus. (Shiokaze and Ishizuchi trains generally split and combine at Utazu station.)
Nearly everybody takes the Ishizuchi Tōzan Ropeway (石鎚登山ロープウェイ)  halfway up, although it's fairly steeply priced at ¥1000/1900 one-way/return. Make sure to start climbing early enough as the last ropeway returns at 6PM.
If you choose to walk instead, be aware the "path" is poorly marked at best and quite dangerous in parts. It will take over two hours. It's quite a hike up from 450 meters to 1300...
For many visitors, the final destination is Jōju (成就), the site of the main shrine grounds — a classically elegant and austere complex notable mostly for the variety of stone hammers on display here and there. The shrine is a leisurely 20-minute walk up from the ropeway station at an altitude of 1450 meters.
The thing to do in these parts is, naturally, to climb up to the summit. Figure on 3 hours for the ascent and at least 2 hours for the descent. There are lockers at the ropeway station if you want to leave excess weight behind, although they are too small for most rucksacks; ask nicely and maybe the staff will stow your gear for you.
Buy, Eat & Drink
There is a small group of souvenir shops and restaurants outside the Joju shrines. Most are open 7AM-7PM. Figure about ¥1000 for a small meal or ¥200 for a bowl of miso. There are also a few restaurants on the main road at the bottom of the ropeway.
Bring plenty of water if you intend to walk up to the ropeway station, and avoid drinking from the mountain streams. Bottled water is very expensive at the top, as is beer — about ¥600 for a can of Asahi.