In reply to Mike Conlon:
Not sure which article you're refering too, maybe this one?
http://www.ukclimbing.com/articles/page.php?id=6372
If you're dead set on Chamonix then the Aiguille Rouges are tailor made for the sort of thing you're on about. I've not done much there but the NE arete of the Perseverance was brilliant and had a fairly alpine feel compared to other routes Ive done in the Aiguilles Rouge which were either v crowded on felt like cragging.
Also from personal experience the Saas Valley is a great place to start for summer alpinism. Routes on the Dri Hornli and Jegihorn are low altitude (relatively), pretty free from objective danger, just above a hut for the former, and easily accessible by lift in the latter. The Portjengrat traverse is a brilliant next step up if those first routes go well, plus there's some fairly low commitment 4000-ers in the valley; the Allalinhorn is a short glacial walk from the high lift, and the Nadelhorn is a straightforward snow climb, albeit with a long approach to the hut.
Others I know have done good routes in the Arolla valley when starting out, plus further East the Bregaglia has some low commitment/danger rock routes to get started on (eg Spazzacaldera)...
To balance out the posters shooting you down for being concerned about objective danger/altitude etc, I'd say those are very legitimate concerns, especially if you have no experience, and it's worth doing some routes to get you used to the environment/need to move fast etc before you venture onto bigger and more committing things.