You can walk most of the old town but for an excursion take the mountain railway to Pleiades for some great views and skiing in season. Make sure you get the Riviera Card at your hotel, you'll be able to use the bus for free.
You can also get a free bike for whole day with CFS 20 deposit, which just in front of "COOP" super market next to the train station.
Musee l'Histoire de Vevey -- from the main train station take the main street, turn left and go along until you'll encounter the signs
Monument de Henryk Sienkiewicz - the famous Nobel-prize winning writer's monument is located in the Grand Hotel du Lac (it's in the garden but the hotel's personnel is happy to help and talk about it). The writer was living and working there during his political emmigration from occupied Poland.
'Mont Pelerin - from the main train station take the Bus 1, direction Vevey-Funiculaire (ticket 2.80CHF). The ticket for the funicular costs 14CHF and offers nice views. The views and atmosphere on the Mont Pelerin are marvelous, but do take the hike trip to it's summit where the telecommunication tower is located. It takes a bit more than an hour to get there, following an asphalt path through the rural area, yet, sometimes it is rather sloppy but nothing special. The elevator for the tower costs 5CHF (3CHF for children <16yr, but recently it sells only children tickets due to a fault) and is fully automatic. It offers magnificent panoramic views on the Alps, Vevey and Montreux.
National Bar Expect the service to be a little slow when it's busy but if you're not in a hurry and are out for a good time the great atmosphere makes the wait worthwhile. The food is great and they keep changing the menu every month. During the summer they have a great terrace and in the winter it's the best place to go. It's located just outside the border of the main square on the right side if you're facing the lake.
Egli Bed and Breakfast, chemin du Petit-Centenaire 8, 1802 Corseaux. +41 21 922 60 87,  They have a cat, so beware if you are allergic. Views of the lake are amazing. You are staying in someone's home though. Nice place to start the wine hike or go up the funicular for dinner.