Are you going to stop on the French side at Chamonix or Italian side at Courmayer? You have a choice of lifts -- if the weather cooperates, it can be quite nice. The surrounding lifts near Chamonix, like Les Houches, take you up in elevation to several great trailheads. At Aosta, there is a cut through road over the Alps to Martigny, if you are driving and don't want to take one of the tunnels. I'd only tackle it in the mid to late summer, as it's really a two lane road, in cloud free weather so you can get the big view. Yes, it's the Great St. Bernard Pass and there are many great hiking choices there. I've stayed in ski-towns like Verbier during the summer as launching points for hikes as well.... there are several nice towns in the Valais that lose their ski-crowds and become quite pleasant. You can also continue up the Valais to Zermatt, of course.
I think the others have you covered on the stays in Vevey, and visits to Montreux and Ouchy (my former 'hood in Lausanne). I prefer Vevey to Montreux. I do recommend the ferry for a day trip, as the view is great. You can also get to Evian and other points along the lake. There is a cog wheel train to the peak at Rocher des Nayes -- there is also a hike that originates in Montreux, or less steep variants in/around the lookout, which offers superb views over Lac Leman. The wine area in this area is call Lavaux, and it's possible to do tastings and to walk along the terraced vineyards. It's best to stick to the whites here. There are even smaller towns here, like charming San Saphorin -- you can do the vineyard walk between Chexbres and San Saphorin.
The famous castle is Chateau du Chillon.
Some of the hikes and refuges have restaurants nearby -- the mountain cheese fondue is a must-not miss.
For bookings, consider searching a mega-search engine like HotelsCombined.com and/or Kayak.com. I'd also look for rentals via booking.com, Flipkey/TripAdvisor or Homeaway.