Going from Vevey to Chillon the bus is on the same side as the lake. So no need to cross the street if you are headed for Chillon. Just walk to the lakeside promenade from the bus stop. Bus number, BTW, is 201 and you can catch it in Vevey in a lot of places including the station. Runs every 10mins or so.
If you want an unusual place to eat in Montreux, try le Palais Oriental which is right on the lakefront, has a lovely veranda, and serves Middle Eastern and Iranian food. In Vevey there is a super little restaurant called Les Negotiants, in the hotel (also nice) of the same name in the old town and an excellent sushi restaurant right across the street, Ichioshi..
Saturday market in vevey is huge! And super!!