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If you just want to travel around the Lac Leman you might want to go to Montreux which is very famous and offers a beautiful view over the lake and the mountains. Some people say it's the most beautiful view over the lake... To really enjoy the view you can actually go to this restaurant which is the wine yards and they offer drinks in a relaxing lounge atmosphere and of course restaurant too.
If you are in Chamonix the first time and if the weather is good you should plan in a trip to go to the top of the Aiguille du Midi. You can take the cable car to go to the top and from there you have an amazing view over the Mont Blanc and the Alps. The cable car departure is in the city center of Chamonix and you will need to switch the cable cars once to go up to the top. Just a hint: Even in summer with 30 degrees in Chamonix you will have temperatures close or below 0 degrees Celsius. So you might want to pack in a pullover ;o) In case the weather is cloudy... don't go up as potentially you don't see anything as you might be just in the clouds.
Chamonix is about 1,5 h drive from Geneva and a great day trip. Also in summer times hotel rooms are relatively cheap. The city center is not big but offers a nice walk through historical houses. This Mere de Glace is a glacier and you can actually walk through the glacier. I believe it's open all year and constantly changing as the ways are carved through the glacier but of course the glacier is moving and therefore it is never the same.