Vevey town and waterfront

Vevey is a youthful town on Lake Geneva with a beautiful walking path on the water. The views of the high mountains from the walking path are magnificent, and there are rows of trees and flower beds. There is a little playground for kids with a tiny pyramid climbie and tiny merry go round. As you walk along, the trees are absolutely gorgeous and so are the buildings. A small Charlie Chaplain statue along the walking path honors the actor who was a longtime resident of Vevey. A giant fork sculpture, La Fourchette, sticks out of the lake! Along the walking path here is the Alimentarium Museum, a food museum run by Nestle, in an elegant mansion where Nestle headquarters used to be. The museum has a lovely garden, and even if you don't go in, it's wonderful to walk around the garden for free and check out the herbs and vegetables. The museum itself is very hot and stuffy, but has some interesting exhibits of food making tools, 1950s food packaging, and other colorful things that kids enjoy. Your hotel gives you a discount pass called Montreux Riviera Card that offers 50% off the admission for these museums and other museums, plus free bus and train rides, and 30% off ferry rides.

There is summer live music on the waterfront at Vevey, at Quai Perdonnet! It is usually Fridays 7-9pm and Sundays 4:30-6pm, June to early September.
Vevey has a much nicer inland downtown area than Montreux, with cute, historic buildings and peaceful streets around Place du Marche, which is actually a parking lot but with lovely buildings around it. A fantastic place for families to eat is Comelon Burrito Bar (closed Sat dinnertime, Sun, and Mon) on Rue des Deux-Marches. It is counter service, and you get to choose what you want in your burrito through the glass. The staff are friendly. We were so happy to find this place since the cafes in the square don't serve dinner until 7pm and we were all hungry way before that! Many smaller towns in France and Switzerland don't serve dinner until 7pm and it can be quite difficult when you have kids low on blood sugar and also ready for bed.
It's easy to take the train to Vevey from Montreux, though it can be crowded at rush hour. There are some sketchy people that wait by the pedestrian underpass tunnel where you must leave and enter the train station. They try to surround you, walking at you from all directions. Hold onto your belongings, and walk quickly, with confidence. The bus is not recommended, as it is too bumpy. It's super fun to take the ferry, which is fast and very scenic! If you have a chance to take it, do! Use the SBB Mobile app to find the time of the next train or ferry departure. If you drive, you can park at Place du Marche and you will be in the center of town, and by the waterfront too.
Two museums we didn't visit, that are also on the hotel discount card:
Near the train station on a main road is Musee Jenisch. It is a very small museum in a lovely neoclassical mansion, with a painting by Courbet and one by Picasso- come when they have a special exhibit. 
If you have a car and are interested in Charlie Chaplin, you can visit Chaplin's World, his house up high on the hill, an eight minute drive away. The hotel discount card gives a discount off the $25 admission. Open daily 10-7.

Directions

Comelon Burrito Bar, 23 Rue des Deux-Marches, Vevey. Closed Sat dinnertime, Sun, and Mon.
Alimentarium Museum, 25 Quai Perdonnet, Vevey. Closed Monday. Open Tues-Sun 10-6. Fee is $15, and $6 for children aged 6-15. You can get half off with the discount card they give you at your hotel. The garden outside is free to enter.
The main square, Place du Marche, is also a large parking lot. It is by the waterfront and all the cute streets and buildings.

Click on map for interactive view