The nearest airports are Geneva (90 km) and Lyon (220 km). Geneva is by far the more convenient option — there are direct buses (Alpybus, Ouibus, and others) from Geneva Airport to Chamonix's bus station that take about 90 minutes and cost around €30–40 each way. Taxis are significantly more expensive (€180–220). A car from Geneva takes about the same time.
By train from Paris the journey takes around 5.5–6 hours with a change in Geneva or Saint-Gervais. The Mont Blanc Express narrow-gauge railway runs from Saint-Gervais to Chamonix and continues into Switzerland — a scenic final leg that is worth doing for its own sake. From the UK, the Eurostar to Paris with an onward TGV to Geneva and then the Mont Blanc Express is a viable, lower-carbon alternative to flying.
Driving into the valley in winter requires snow chains or winter tyres — the road from Sallanches climbs steeply and conditions can be severe. The Mont Blanc Tunnel connects Chamonix with Courmayeur in Italy (15 km, about 20 minutes) and is a busy freight and tourist route; a toll applies.