In reply to MalcyversustheMunros:
This should answer most of your questions: www.chamonix.net/english/leisure
Highlights include: mountain biking, rafting, canyoning, valley rock climbing, visiting glaciers, great wildlife areas, mountain walking ( with uplift to take out the sting for the less enthusiastic ), cable cars offering amazing views and, of course, Tartiflette.
For the climbing / guiding side this should help: www.chamonix.com/high-mountain-office,4-492959,en.html
In particular you can get up to date information about walks, huts, climbs from this site.
Here is a link for the highly recommended British Mountain Guides: www.bmg.org.uk/activities/
Whilst there are plenty of activities and some awesome walking May June is quite early in the season. It varies every year but some of the higher walking routes are likely to have patches of snow in sheltered / shady areas that can be bullet hard first thing in the morning. Not unusual to need crampons and / or axes for some sections.
Routes you might consider with a guide could include:
Crochues Traverse
Aiguille Tour area has several nice peaks as someone has mentioned
Traverse of the Vallee Blanche
There are a couple of short easier peaks accessed from the Torino Hut area ( Italian side ).
Petit Mt Blanc.
Have fun,