I think your hotel should work well for you. It's centrally located and will allow you to take the bus at your doorstop and the train with access via the lift or the escalator to the train almost next door. I rarely use my car and take the same bus and train you will take to Vevey. Across the street from your hotel is the Marche in Montreux with a stop for the boat. The tourist office is across the street to the right a few minute walk away. If you are feeling up for a walk, take a left at the water and walk along the waterfront path to Chateau Chillon.
If you wanted to walk part of the vineyards, I would take the train from Montreux to Chexbres or Grandvaux at the top of the vineyards so you can walk down rather than up. I have seen signs for an app that you can download on your phone to guide you, but you can also ask for the easy walking map at the tourist office.
If you are up for a short walk from the Marche in Vevey, take a left at the water and walk along the waterfront towards the chateau of La Tour de Peilz, a very charming port village. You can easily walk out on the jetty for a beautiful view back towards La Tour and Vevey. There's a cafe in the courtyard of the chateau for a coffee, drink or a meal. It is run by a friendly family who speak some English.
Switzerland is expensive to visit. To save on dining out, I recommend a local place in Montreux called Cafe-Restaurant du Globe which is a five minute walk from your hotel on Rue de l'Eglise. I had never been before, but a friend on a budget brought me there once. It's very basic and I did not hear any English on my visit, but they had roasted chicken and fondue on their menu which were almost half of the price you would find in other places.
At the ground level entrance of the Migros supermarket at the small shopping center opposite your hotel, there's a cafe to the right of the entrance which offers buffet style prepared foods which are well priced and they offer seating.
Another good option is in Vevey at the Manora Restaurant on the top floor of the Manor department store, located just the opposite corner from the train station. It's set up buffet style, the food is very good, and the prices are reasonable.
If you decide to go to the Marche in Vevey on Tuesday or Saturday morning, visit the vendor near the large pillared building at the top of the Marche which houses the tourist office. They sell individual warm cheese quiche for 3chf each which draw long lines but are worth it. There's a nice cafe the opposite side of Marche with outdoor seating where you can have a coffee or fresh pressed juice.
All of the supermarkets sell prepared foods you can take with you.