By train is the most practical option. Vichy has its own station on the main Paris–Clermont-Ferrand line, with direct TGV connections to Paris Gare de Lyon taking around three hours. From Clermont-Ferrand (the regional hub, 45 minutes by regional train) there are connections to Lyon and further south. The station is about ten minutes' walk from the Parc des Sources.
By car, Vichy is accessible from the A71 autoroute (Paris–Clermont-Ferrand), leaving at the Vichy exit. Driving time from Paris is around three and a half hours. There is adequate car parking near the centre, though the streets around the thermal district can be busy in high season.
By air, the nearest airports are Clermont-Ferrand Auvergne (about 55 km, with car hire available) and Lyon-Saint Exupéry (about 150 km, with good motorway connections). Neither has extensive international routes, so most visitors flying from outside France will connect through Paris.