Nine days from Geneva to Turin: over Mont Blanc, into the Italy nobody queues for

Geneva's old town climbs a small hill on the left bank and everything on it is within ten minutes of everything else. Start at the top, in the Place du Bourg-de-Four, which has been a market square since the Romans and now holds about eight terraces under plane trees.

Saint-Pierre is the reason the hill matters. Calvin preached here from 1541 until his death in 1564, and the Reformation stripped the interior back to bare stone and clear glass, which is a shock if you have come from a French cathedral. His chair is still in the nave and it is not much of a chair. Under the floor there is an excavation of everything that stood on the site before, from an Allobrogian sanctuary to two fourth century basilicas, and it is far better than it sounds. The north tower is a few flights of worn stone and gives you the roofs, the lake and the Jura in one turn.

Finish at the water. The Jet d'Eau began life in 1886 as a pressure valve for the city's hydraulic power network and was so obviously spectacular that the city moved it into the lake and kept it. It throws water 140 metres up and the spray lands on whoever is standing on the jetty, which is the point.

North of the station, on a hill given to the League of Nations by the city in 1929, sits the second largest United Nations complex in the world. The Palais des Nations was finished in 1936 for an organisation that had already failed to stop what was coming, and the building went straight on to house its successor. The guided tour takes you into the Assembly Hall and the Human Rights room with the melted-looking coloured ceiling, and you need your passport at the gate.

Ten minutes downhill is the museum of the Red Cross, which was founded in Geneva in 1863 after a Swiss businessman walked across the battlefield at Solferino and could not forget it. The museum is arranged around three ideas rather than three centuries, and the wall of prisoner of war index cards, millions of them written out by hand, does more than any caption could.

Then stop reading and get in the water. The Bains des Paquis is a concrete pier from 1932 sticking out into the lake, open to anyone, with a sauna, a hammam and a canteen that serves the cheapest good plate in Geneva and a fondue in winter that people cross the city for.

The train leaves Cornavin, runs along the French border and changes at Saint-Gervais onto a narrow gauge line that grinds up the Arve valley with the massif filling the left-hand windows. It takes most of the morning and it is not a chore.

Chamonix itself is a long thin town squeezed between two walls of rock, and the first thing to do is look up. The statue by the church shows Jacques Balmat pointing out the summit to Horace-Benedict de Saussure, the Genevan scientist who put up the prize money that got Mont Blanc climbed in the first place. Balmat and Michel Paccard collected it in 1786. Saussure went up himself the year after with eighteen guides.

The Alpine Museum, in a grand old hotel on the main street, tells the rest: the guides' company founded in 1821, the crystal hunters who knew the mountain before the climbers did, the first winter Olympics held here in 1924.

The cable car starts in the middle of town at 1,035 metres and finishes at 3,842, a vertical gain few cable cars anywhere match, and it does it in two stages with a change at Plan de l'Aiguille. The upper stage is a single unsupported span so long it feels like a mistake. At the top there is a bridge between two rock towers, a tunnel through the summit, a glass box you can stand in over a thousand metres of nothing, and Mont Blanc close enough to see the tracks of the people walking up it.

Go on the first cars of the day. The air is clearest before ten, the queue trebles by eleven, and the cloud that sits on the summit most afternoons has not formed yet. It is twenty degrees colder up there than in the town and there is a third less oxygen, so take the coat you were not going to bring and move slowly.

In the late afternoon cross the valley and go up the other side. The lift on the north wall climbs into the sun, and that side is the only place to see the whole massif at once: the Aiguille you were standing on this morning, the Mer de Glace, the summit dome, all of it lit from the west.

The rack railway to Montenvers opened in 1908 and takes twenty minutes to climb to a hotel and a terrace above the biggest glacier in France. The Mer de Glace runs 7.5 km down from the slopes of the Grandes Jorasses, or twelve if you count what feeds it, and it moves about a centimetre an hour, which is fast for ice.

It is also going. The staircase from the station down to the ice cave is rebuilt and extended most years as the surface drops. Panels beside the steps mark where the surface stood decade by decade, and you pass every one of them on the way down. Whatever you think about the subject in the abstract, the walk down makes the argument.

In the afternoon take the gondola from Les Praz up to La Flegere at 1,877 metres, which faces the massif square on. There is a chairlift above it to the Index at 2,595, and a level path in either direction along the valley wall. This is the walking half of the trip and the reason three nights is better than two.

The road out of Chamonix climbs to the tunnel mouth at Les Pelerins and goes under the mountain for 11.6 kilometres. It opened in 1965 and cut the drive from Chamonix to Courmayeur from hours over a mountain pass to a quarter of an hour. You come out in Italy, in sunshine more often than not, with the same massif behind you looking completely different.

The Aosta valley below is worth the detour it does not really require. Castles sit on crags the whole way down, the Romans left a triumphal arch and a theatre at Aosta itself, and the roadside restaurants do polenta, fontina and cured beef. Take two hours over lunch here and you still reach Turin before the evening.

Turin arrives as a grid. Get the car parked, then walk into Piazza Castello and out under the arcades. The Galleria Subalpina, opened in 1874 off Piazza Carlo Alberto, is glass, iron and cream stucco, holds a cinema and a cafe, and takes four minutes to walk through, which is the correct introduction to a city that spent two centuries on its street furniture.

The Museo Egizio opened in 1824, which makes it the oldest museum in the world devoted to Egypt alone, and only Cairo holds more. It exists because the Savoy kings bought a French consul's collection wholesale and then funded excavations for a century. The rock-cut temple of Ellesiya is in here, moved stone by stone when the Aswan dam flooded its valley. So is the tomb of Kha and Merit, an architect and his wife, found sealed in 1906 with the furniture, the food and the linen still in it. Three hours, and book the slot.

In the afternoon walk north to Piazza Castello and into the royal palace, seat of the House of Savoy from the sixteenth century and of the Kingdom of Italy after 1861. The Royal Armoury inside it is one of the great collections of arms in Europe, laid out in a gallery a hundred metres long.

The cathedral is next door, plain late fifteenth century brick with a chapel behind the altar that is anything but. Guarino Guarini's black marble dome, built for the Shroud, burned in 1997 and took twenty one years to restore. The Shroud itself is almost never shown; the cathedral holds it in a sealed case and displays it perhaps once a decade.

Porta Palazzo claims to be the largest open air market in Europe and nobody serious disputes it. It runs six mornings a week across a square and four covered halls, sells everything from Piedmontese cheese to Moroccan spice, and is at its best and busiest before ten. The Balon flea market runs alongside it on Saturdays.

In the afternoon, the building on the skyline. The Mole Antonelliana was begun in 1863 as a synagogue, ran so far over budget that the Jewish community gave up and swapped it for a plot elsewhere, and was finished by the city in 1889 at 167 metres, briefly the tallest brick building in the world. Since 2000 it has held the National Cinema Museum, which spirals up the inside of the dome on ramps and beds you can lie on, and a glass lift runs straight up the middle of the void to the viewing deck.

End on the far bank. Santa Maria di Monte dei Cappuccini stands on a bluff above the river, took seventy years to build because of plague and money, and gives you the whole grid of the city with the Alps standing behind it. Go at sunset, when the wall of mountains runs the full width of the view and the arcades below are filling up.

Turin takes coffee more seriously than anywhere else in Italy, and the bicerin is the local proof: espresso, drinking chocolate and cream in a small glass, poured in layers and not stirred. The cafe on Piazza della Consolata has been making it since 1763 and has never seen a reason to change.

Ten minutes north are two brick towers and a double archway, the Porta Palatina, which is one of the best preserved Roman city gates anywhere and was the north entrance to Augusta Taurinorum. The grid you have been walking all week is the grid the Romans laid out around it. The archaeological park beside it is open and free, and it is a good last thing to stand in before a train.