Chamonix is about 75 km to the south east of Geneva. The town itself is very close to the borders of France, Italy and Switzerland. In fact if you stand at the summit of the Mont Dolent, you can say you are in 3 countries at once.
The town is just over 1,000 metres above sea level, so don't expect total trail access after the first big Alpine snowfalls (as early as October). From the town itself, there's very little immediate gravity assisted riding - it's all up on the valley sides and peaks that loom everywhere. If you don't like riding uphill, no matter - there's an extensive lift system to take you to the tops.
Most of the lifts will take bikes, but many of them close after the winter season for maintenance, re-opening mid June through to late September. Prices for one way trips are between £3 and £9 (bikes go free). For 2007 there's a new "VTT" lift pass for mountain bikers which will be about £55 for a week.