The Chemin des Muletiers (french for "path of the mule drivers") is the most obvious way to reach the summit of the Puy de Dôme on foot. It's just a steep path departing from the parking area of the Col de Ceyssat, a trailhead where a couple of bar-restaurants are found. The path starts between both bars, climbing eastwards to reach the summit in 40-45 minutes. No difficulties to be met, just walk on and enjoy the sights when you reach the summit plateau.
The obvious place to enter the region is Clermont Ferrand, the capital city of the Department. It's got fine communications, never mind if you come by train, plane or in your own car. Once you get there, only 13 km by road will stand between you and the summit (by road, that's it). But here in SP we are supposed to like it the hard way, aren't we? Therefore, count about 16 km to the Col de Ceyssat trailhead. You will find a rather big parking area there, together with a couple of bars where food and drinks are available.
The route from Clermont Ferrand to the Col de Ceyssat is the D941. Leave the town center via the Raymond Bergougnan avenue and get going until the small village (almost a peripheral quarter of Clermont Ferrand) of La Baraque. That's where you will have to follow the signposts towards Bordeaux (road D942). Turn right towards the Col de Ceyssat after an additional 3 km or so, the crossroads is properly marked towards "Puy de Dôme", and take care to turn left in the only crossroads left after that. That way you will avoid the summit road and reach the trailhead of the Chemin des Muletiers
A steep hike. Those three words make a fine summary for the climb between the Col de Ceyssat and the Puy de Dôme. Just take the path at its lower end, between the bars and by a II World War small memorial (see picture to the left), and start walking. The path is wide and well cared for, the floor is covered by fine gravel and there are small benches along the way to take a seat if you need a rest. A normal time to reach the summit would be somewhere around 45 minutes, but take your time. There's no hurry.
The climb is a short one. If necessary, you might get to the summit and back down in scarcely 1.5-2 hours. Therefore, nothing is really "essential". If anything, good walking shoes are nice (avoid beach sandals or anything similar...) and a pair of trekking poles could be handy to offer your knees some rest on the way down. Waterproof clothes would also be nice if the weather is rainy, but even that is not really essential, as far as you leave some dry clothing in the car. That's all.