Instead of taking the bus from Montreux to Chillon Castle, it's much nicer to walk there on the lovely lakeside promenade.
The path is always right on the lakeshore, with beautiful fantastic views of Lake Geneva. The path is lined with many sculptures, including the popular circle bench photo spot.
From the Montreux train station, it takes about 50 mins to reach the Chillon Castle. See trail map.
If you want a longer walk, the full path stretches from Port du Basset, at the north edge of Montreux, to Port de Villeneuve, south of Chillon Castle.