The Auvergne region surrounding Vichy is one of the most dramatically underrated landscapes in France — a chain of extinct volcanoes (the Chaîne des Puys), river gorges, medieval villages, and vast empty uplands. Vichy makes a decent base for exploring it.
Clermont-Ferrand (45 minutes by train) is the regional capital, with a striking black-lava cathedral and good access to the Chaîne des Puys volcanic park. The Puy de Dôme, the most accessible of the volcanoes, can be reached by a rack railway from a point near the city.
The Gorges de l'Allier, the deep river canyon south of Vichy, are accessible by car and offer dramatic scenery along a route that runs south through Issoire and into the volcanic uplands. The train line through the gorge is itself an engineering achievement worth knowing about.
Moulins (an hour north by train) is the old capital of the Bourbonnais region, with an exceptional collection of medieval art in its cathedral treasury and a good museum. It is quiet and often overlooked, which is part of the appeal.
Cusset, just 3 km east of Vichy, is a small town with a medieval core quite different from Vichy's Belle Époque character. The Église Saint-Saturnin has notable Romanesque features. It makes a reasonable short excursion on foot or by bike along the river.