You can always take the bus that goes from Vevey to Montreux and Villeneuve - clean, regular and no problem with a pushchair. Get off at Montreux and take the lakeshore walk from Montreux to Villeneuve past the Chateau de Chillon. This walk is an uninterrupted lakeshore path (unlike Vevey to Montreux) and is more varied and longer than the lakeshore promenade at Vevey. You can stop and feed the ducks/swans along the way. It's also a safe place for kids to ride their bikes when they get a little older. When you get to Villeneuve you can catch the bus back to Vevey so you don't have to double back on yourself. If you don't want to go all the way to Villeneuve you can stop at Territet or Chillon and catch the bus back from there.