I've done Haute Route twice, and it's a stunning hike. However, the entire route takes close to two weeks - 12 stages plus two days in reserve for the inevitable bad weather.
The good news are that even five or six days can give you an unforgettable experience if you hike the most dramatic and scenic section of the route.
What you'll want to do is to start from the ski resort Verbier. Simply take the train from Zurich or Geneva to Martigny, and then a short hop on a local train to Le Chable, and finally by gondola to Verbier, from where it's a one hour beautiful hike to your first night - and starting starting point of your Haute Route, the Cabane du Mont Fort hut.
The next five days will take you through the wildest, most demanding and adventuresome part of the Haute Route: Crossing a glacier, scaling vertical ladders up a rock face, and staying in huts with a view to a huge ice fall or stunning north face.
Assuming the weather plays nice five stages should take you as far as the small ski resort of Zinal, from where it's easy to catch a bus to a train station in the Rhone Valley. The first stage is from Cabane du Mont Fort to Cabane Prafleuri (a long and tiring walk, but stunningly beautiful), your next to Cabane Dix (you could quite easily walk straight on, but don't - the hut and the location is great). Third stage takes you across a glacier (short and marked route), up those ladders for a brief view of the Matterhorn from the pass, and then down to the village of Arolla. The official fourth stage is a mare transport stage that you'll want to skip. Take a local bus instead down to Les Hauderes. Your fourth stage is from here to Cabane Moury, dramatically situated on top of a huge icefall. Fifth stage takes you to Zinal.
The weather is late August/early September is the best and most stable of the entire season. Nevertheless, you should add in an extra day for bad weather. We once snowed in.
For the same reason, it makes no sense to book the huts several weeks in advance. Generally you should just do it early the same morning. On Fridays and Saturdays, when many Swiss head for the most popular huts, it may make sense to book the day before if the weather forecast is good.
For a great description of the hike you should get Kev Reynolds' guide, published by Cicerone in the UK.
Have a great hike!