The obvious highlight is the cable car up to the Aiguille du Midi. At approximately 13,500 feet, it is one of the highest peaks in the Mont Blanc Massive, and you can take the cable car up to it (this being the highest cable car in the world). From the top the views are stunning, from the Mont Blanc to the North, to the Aiguilles Rouge towards the south and the Glacier below. In the Midi station there's a restaurant and indoor and outdoor viewing areas, though it's very high and a bit chilly, so wrap up and be prepared to feel the altitude (for most it's just tiredness, and occasionally a bit of dizzyness).
Another option is to visit the Lac Blanc, a lake at about 9000ft, which despite it's name is a vivid rich blue. You can walk a path to it, or take the cable car to the Index station from La Praz golf course, and meet the path there, from which it's just a gentle stroll. It offers one of the best alpine panoramas in the world, from the Mont Blanc Massif to the Grands Jorasses. You'd have to be there to truly appreciate it.
Quick Tips:
As well as the mountains there's also an ice-rink in Chamonix, open all year, and a full olympic sized swimming pool, though lads, I'm afraid they've got a compulsory Speedo's policy. You can rent them at the pool if you want but... well..., no.
Chamonix is a very walkable town, and theres an excellent bus service going from Chamonix through the valley, stopping off at all the little towns along the way. Chamonix is easy to get to, there's a train from Lyon and a bus from Geneva, the latter taking just 45 minutes. You may be asked for your passport as you cross the border (the French can be quite finicky about this) so be prepared.
Best Way To Get Around:
As I said, you can walk around Chamonix easily. With regard to hiking, you can get a bus to most destinations, and there are paths going up most routes, with cable cars for those who are willing. The cable cars can be a little expensive, but otherwise transport is cheap. There's a plentiful supply of restaurants and bars, and the Cyber Cafe on Rue des Moulins (behind the main street) offers an olde worlde English bar for those who want to sit down on a couch and relax.
There's tons more stuff to do, it's a great holiday if you're young and want adventure.