The Route des Grandes Alpes is the longest continuous good driving road in the Alps. From Thonon-les-Bains on Lake Geneva to Menton on the Mediterranean, it strings together seventeen passes across 720 kilometres, and it earns the superlatives attached to it. The Galibier at 2,642m, the Iseran at 2,770m (the highest paved pass in the Alps), the Izoard with its moonscape Casse Déserte, and the descent through the Alpes-Maritimes into Nice, there is nothing comparable in scope or variety in European driving.
Seven days is the minimum civilised pace. Done in five, it becomes a transit exercise. Done in ten, you have time for side trips (the Col de la Madeleine, the Col de la Bonette) that are worth the detour. The route below is the seven-day version, long but not punishing, with nights in towns that have decent restaurants and garages.
Thonon is the start. Flying into Geneva, collect the car, drive forty minutes east along Lake Geneva, and you're at the first pass (Col des Aravis) within the hour. The ending city is technically Menton, but Nice is where you return the car, thirty minutes along the coast from Menton, and with a proper airport for the flight home. Plan the rental accordingly; most major operators allow Geneva-pickup/Nice-return for a fee.
The passes break naturally into three thirds. The northern third (Aravis, Saisies, Cormet de Roselend, Iseran) is classic Savoyard driving, wide valleys, dark-green forests, high meadows at altitude. The central third (Galibier, Lautaret, Izoard, Vars) is the dramatic core, wide high passes in the Dauphiné Alps with Tour de France history written all over them. The southern third (Cayolle, Champs, Valberg, Turini) drops into the Alpes-Maritimes, smaller, twistier, Mediterranean light, the road becoming a Monte Carlo Rally stage as it nears the coast.
French alpine passes are open roughly June through mid-October. The Iseran and Galibier typically open latest (mid-June) and close earliest (early October); plan accordingly. Fuel is widely available. The autoroutes are tolled and expensive, but you should avoid them anyway, the whole point is the back roads. Surface quality on the major cols is good; some departmental roads in the south are patched. Traffic is lightest midweek and early in the morning; French school holidays in July/August bring motorcycle traffic on every col.