Chamonix has changed over recent years - either you want to be part of the huge 'Brit Pack' or try to learn and feel the mountains. Whether alone or with family I most strongly recommend (for the price of a few tanks of petrol) hiring a fully qualified French mountain guide (or Aspirant) for a few hours, who will also be a ski instructor. He/she will know exactly where to go and when, get priorities on transport and above all, take as much time as necessary to make you feel and understand the mountains. Their enthusiasm and knowledge is infectious!
If you enjoy each other's company you can probably exchange phone numbers - I skied with a guide and his son by invitation on a Sat morning for a nominal fee. Instead of instruction, my wife was taken to see a recent avalanche and cross and see some snow bridges.
To ski Vallee Blanche in a collective group is just a rush to get the guide home as soon as possible - with your own guide the day is yours and with good relationship you can be taken far away from the crowd in safety in an incredible landscape.
I just give a general warning, that Chamonix's piste gradings (blue/red/black) can be much more demanding than elsewhere!
Bureaux des Guides (Chamonix or Houches)