Vichy's thermal district is compact and walkable — the Parc des Sources, the opera, the casino, the main spa buildings, and the WWII historical sites all lie within a few hundred metres of each other. Most visitors staying in the centre will not need any transport to see the main sights.
Cycling is well catered for. The Via Allier cycling trail runs along the river, and the flat terrain of the Allier valley makes it genuinely easy riding. Bike hire is available from several points in the city, and the lakeside parks south of the centre are ideal for a traffic-free afternoon ride.
Bus services cover the wider urban area and reach the surrounding communes (including Bellerive-sur-Allier across the river). For getting between the city centre and the train station, buses run regularly, though the walk (around 15 minutes) is not unreasonable.
Taxis are available and can be useful for reaching restaurants or sights slightly further from the centre, or for getting to and from the train station with luggage.
By car, the central thermal district has narrow streets and limited parking. If you have a car, park it near the station or in one of the peripheral car parks and explore the centre on foot.