Central Vevey is easily walkable. The lakefront, the Grande Place, the old town, and the main museums are all within 15 minutes on foot of the railway station.
For Chaplin's World, take bus 212 from the station — it runs to the museum gates and takes about 10 minutes. For Villa Le Lac in Corseaux, either walk 2 km west along the lake or catch a CGN steamer to the Corseaux landing stop.
The Mont Pelerin funicular departs from a station a short walk above the town centre and reaches the ridge in about 25 minutes. Swiss Travel Pass holders ride free.
Bikes can be rented at the station through the Swiss national SBB scheme. The flat lakefront path east to La Tour-de-Peilz and Montreux is the obvious cycling route. The CGN paddle-steamers offer a leisurely way to reach Lausanne, Montreux, or Chillon without retracing the train route.