Ten days in Switzerland: the trains go everywhere, including straight up

The airport train takes ten minutes and puts you under the Hauptbahnhof, which is five minutes' walk from everything below. Drop the bags and climb the Lindenhof, a gravel terrace under lime trees where the Romans had a fort and Zurich now plays chess. From up here the whole layout makes sense: river down the middle, old town either bank, lake at the end of it.

Cross to the Grossmünster. It is a plain Romanesque church that changed European history: Huldrych Zwingli was made people's priest here in 1519 and the Swiss Reformation followed, which is also why the interior is so bare. The tower stairs are narrow and the view down onto the roofs is worth them.

Eat in the Zeughauskeller, an arsenal of 1487 turned beer hall, at a long shared table. It is touristy and the portions are ridiculous and it is still the right first night.

Take the S10 from inside the main station up the Uetliberg. It takes about twenty minutes and lands you on a wooded ridge with a viewing tower on top, the lake below and the Alps along the horizon on a clear day. The Planetenweg runs off along the ridge, twelve kilometres with the planets set out to scale, and you can walk as much of it as you fancy before dropping back to a station.

Back in town, the Fraumünster is a former convent church with a set of choir windows Marc Chagall designed in his eighties. No photographs are allowed, which means everybody actually looks at them.

Finish in the river. The Letten baths are fifteen minutes upstream on foot: get in at the top, let the Limmat carry you down to the steps, climb out and walk back up. That is what the city does with a hot afternoon, and doing it once is worth more than another museum.

The train to Lucerne takes about three quarters of an hour and runs twice an hour, so there is no need to be up early. The station in Lucerne sits on the lake with the boat piers outside the door, and the old town is five minutes over the bridge.

That bridge is the reason people come: a covered wooden crossing put up in the fourteenth century, running at an angle across the Reuss, with painted panels in the rafters. A fire in August 1993 took most of them, and you can still see which are old and which are replacements.

Spend the afternoon on the Museggmauer, the rampart wall of 1386 along the ridge above the roofs. Four towers are open, the wall walk connects them, and the Männliturm lets you out onto its roof. Then eat under the town hall, in cellars where the beer is brewed on site and the chickens turn by the door.

This is the first big mountain day and it starts on the water. An old paddle steamer leaves from the pier outside the station and takes about ninety minutes to Alpnachstad, bending between cliffs and changing character every few miles.

From the pier the cogwheel railway climbs to the top of Pilatus at a gradient no other rack railway in the world matches. Up there you get terraces, a ridge walk and, in clear air, an arc of Alps that makes the whole trip make sense. Give it three hours even if you do not intend to walk anywhere.

Come down by cable car towards Kriens and take the trolleybus back into town. If the legs still work, the wounded lion carved into a cliff for the Swiss Guards killed in Paris in 1792 is ten minutes uphill from the old town and best at the end of the day, when the coaches have gone.

Today's transfer is the best ride of the trip. The narrow-gauge line to Interlaken runs along Lake Lucerne, then cogs its way over the pass at Brünig and drops down the far side past Lake Brienz, and it takes about two hours end to end. Sit on the left going out.

Interlaken itself is a resort town on a flat strip between two lakes: hotels, watch shops, hikers and a very large meadow in the middle that was deliberately never built on. Walk out onto it and look south. The gap in the green hills with snow behind it is where the next two days happen.

Eat early on the meadow, at the old café restaurant that has been feeding arrivals since they came by paddle steamer, and get the timetables straight for the morning.

Take an early train up to Lauterbrunnen. The valley is a trench with a flat green floor and vertical walls, and water comes off the top of them in long ribbons: seventy-two waterfalls, by the local count. Staubbach, behind the church, drops nearly three hundred metres in one go and turns to smoke in the wind. A path takes you up behind it.

A few kilometres further up the valley, Trümmelbach is the strange one. Ten more falls, but inside the mountain, carrying meltwater off the three big peaks at twenty thousand litres a second, with a lift cut into the rock to reach the galleries alongside them.

In the afternoon go up the other side. The cable car from Stechelberg reaches Gimmelwald, a shelf of grass with farms, hay barns, no road and a view straight across at the wall of rock. Sit outside somewhere with a drink and watch the light move on it.

The other branch of the railway ends at Grindelwald, in a bowl of pasture under the north face. From the village a long gondola climbs to First, a ridge above two thousand metres with a steel walkway bolted along the cliff below the top station, hanging out over a lot of nothing.

The paths up there are gentle and well made. An hour on foot gets you to a lake that holds the reflection of the whole range on a still morning, and you can keep walking downhill afterwards for as long as your knees agree.

In the afternoon, take the yellow postbus up the road east of the village to Grosse Scheidegg: a grass saddle at nearly two thousand metres with cowbells, an inn and the rock wall at your shoulder. Ride up, walk down, and check the timetable before you set off, because the buses are hourly at best.

The run to Geneva takes about three hours with a change at Bern, along the lakes and through the vineyards of the Vaud. It is the longest leg of the trip and still shorter than most European drives.

Cornavin station is fifteen minutes' walk from the old town, which is a steep grey knot of lanes on a hill: the city hall with its square of cannons, Rousseau's birthplace, antique shop windows worth stopping at.

The cathedral at the top is where Calvin preached from 1541, and where a religious experiment strict enough to be exported across Europe was run. Underneath it is an archaeological site that explains why anybody built a city on this spot; above it are towers with the lake, the roofs and, on a clear evening, Mont Blanc.

Tram 15 runs north to the international quarter, where a dozen organisations sit in gardens above the lake. Book the Palais des Nations ahead: tours only, a few a day, and a passport check at the Appia gate. It was built for the League of Nations, and the tour takes in the Assembly Hall and a great deal of donated art.

The Red Cross museum is a short walk up the road, in the basement of the headquarters. The subject is the most serious in the city and the presentation divides people: it is a designed experience of screens and audio rather than a set of cases. Go for the archive material.

Then come back down and get in the water. The Bains des Paquis runs out into the harbour on a stone jetty: diving boards and a beach in summer, sauna and fondue in winter, and a cheap plat du jour all year. The fountain is straight across the water from the end of it.

One short morning left. The Parc des Bastions sits between the old town and the university, with outdoor chess under the trees and the Reformation Wall down one side: Calvin, Farel, Beza and Knox carved several times life size, watching over a city that stopped listening centuries ago.

Walk down to the water and along to the Jet d'Eau. It started as a pressure release valve on the hydro system, the city liked the look of it, and since 1891 it has been a permanent fountain throwing a hundred and forty metres of lake into the sky. There is a pier out to the base. Taking it means arriving wet.

Cornavin is fifteen minutes back along the quay, and the airport is six minutes from there by train, so this morning works even with an early afternoon flight.