With Swiss trains and buses you can easily get by and you won’t need a car. I would check with the hotel as soon as you get there and ask their advice about any possibilities for getting some kind of pass: for example is there a pass for unlimited train / bus rides for one week. If there is such a pass, it would very likely be cheaper than buying a train ticket every day.
From Montreux you have a beautiful view of Lake Leman and the hills on the opposite side. Remember that when you are looking across the lake, you are actually seeing France. If you want to say that you were in France while you were there, you could take the boat across the lake to one of the cities in France on the lake - but the French towns on the lake are nothing spectacular. To go from Montreux to Geneva you could take the train or the boat. I personally would take the boat one way (from Montreux to Geneva) and the train back, or, depending on the boat schedule, the boat part of the way (e.g. Montreux to Vevey or Lausanne) and the train from Lausanne to Geneva and back from Geneva to Montreux. There is also a nice pedestrian walkway that goes on the edge of the lake from Montreux to Vevey or maybe even all the way to Lausanne. Take a short walk on the lakefront (or a longer one if you are a good walker.)
Assuming you are limited to day trips because you want to be back in Montreux every evening to sleep in the hotel, I would really recommend staying in Switzerland. Going by train outside of the country will take several hours. So unless you want to leave at 6 in the morning and be back at midnight, your day would be (for example) 4 hours to get to city X, 1 hour there, 4 hours to get back. Remember that aside from the trip time as shown by the train schedule, you then have to get from the train station to the sights in whatever city, and then get back to the station with some spare time so that you’re sure not to miss the train. I personally recommend shorter trips and staying in Switzerand. Assuming you don’t have a car, I would also recommend limiting your visits to cities.
What you should see:
You are close to Geneva so you should definitely go see the city and its famous water fountain in the lake, and the flower clock in the Jardin anglais. You will want to take the time to get to the Promenade des Bastions (walk or take the bus, but it’s not far) to see the famous Mur des Réformateurs. Also the Jarin des Alpes with the monument Brunswick is pretty. Remember that parcs in european cities will be small compared to the parks that you might be used to in American cities. Also that commonly people walk on the paths and don’t walk on the grass.
One day go to Lucerne. I know it’s cliché and every tour bus in the world goes there, but it is a very pretty city. Walk around the old town and its small quaint pedestrian streets. Another reason why Lucerne is a must-see is because, a short bus ride from the station, you can go to the base of Mt. Pilatus and take the cable car to the top. If I remember right, it is a three cable-car ride to the top of the mountain (meaning you take one, change to another, change to yet another.) You can’t go to Switzerland without climbing at least one mountain. The cable car rides are impressive (I don’t like heights and I’m always nervous about being suspended from a small cable so high up in the air.) And Mt. Pilatus is the place of legends: it is called Mt. Pilatus because Pontius Pilate, overcome with grief after having sentenced Jesus Christ to death, went to Switzerland and drowned himself in a lake on Mt. Pilatus and there are recorded cases of local farmers having seen dragons flying over the countryside from their perches on Mt. Pilatus. Regardless of what the skeptics would have you believe, these are the true facts behind Mt. Pilatus.
If you like shopping (and you have money to throw away, and you don’t mind that the dollar now is worthless, and that Switzerland is expensive), then take the train ride to Zurich (the New York of Switzerland) and stroll down Bahnhofstrasse to look at the shops.
Keep in mind that Switzerland is a multi-lingual country. In Montreux and Geneva everyone speaks French and signs will be in French. In Lucerne and Zurich everyone speaks German and the signs will be in German. If you need help, every big train station should have a Tourism desk (Office du Tourisme or look for a sign with a lowercase i in a blue circle) - go ask them.
If you decide to get a car for one day, I would have different recommendations. Let me know.
Starbucks is making its appearance in Switzerland. When I walked out of the train station in Lucerne, the first thing that greeted my eyes was a Starbucks. You can, with some due diligence, probably find American fast foods. But as other people have said, dining in Switzerland will be expensive. If you want to eat cheap, I would go inside a Migros (the biggest chain of Swiss grocery / department stores). Look for the MMM migros (M = small store, MM = medium store, MMM = large store) which usually have a cafeteria. In Europe people eat at certain times and restaurants expect to serve breakfast in the morning, lunch from 12 to 2, dinner in the evening from 6:30 or 7:00 on. Don’t be surprised if a small restaurant will actually be closed outside of those hours. I still remember the look on my wife’s face when we were at the Migros self-serve cafeteria at 2:00 and she asked the chef for the last roasted chicken plate. The man said “Non!” and took it away back to the kitchen. It was 2:00 and lunchtime was over. In restaurants, don’t look for the same variety in the menu as you would expect in an American restaurant. If they serve bread with the meal, it will likely be without butter. In some restaurants, if you order a glass of water, they might bring you a bottle of water and charge you for it. If you order a cup of coffee, it will not be a Starbucks-size venti; it will be a “normal” cup of coffee which will probably seem small to you. Don’t expect free refills on drinks!
the dollar will still be on parity with the Swiss franc when you go there and it will be easy to compare prices - CHFr 20 = $20. (CH is the abbreviation for Switzerland: cite