Ten days from Strasbourg to Bern: choucroute in France, and a capital you can swim through

Strasbourg's cathedral is built from pink Vosges sandstone and it does not look like anything else. Work ran from 1176 to 1439 and the single north spire finished at 142 metres, which made it the tallest building in the world for over two hundred years and still makes it the tallest cathedral tower in France. Only one spire was ever built, which is why the front looks lopsided in a way you cannot unsee.

Inside, the astronomical clock is the thing everybody waits for. The current mechanism dates from 1842 and drives a planetary dial, a perpetual calendar and a procession of apostles that runs at half past twelve, which is why the queue starts at noon. Go up to the tower platform too, a little over three hundred steps, for the whole plain and the Black Forest beyond it.

Facing the cathedral square is a merchant's house whose carved upper floors went up in 1589 and cover almost every centimetre of wood with figures. It is a restaurant now. Eat somewhere else, in a winstub, which is the Alsatian answer to the bistro: dark panelling, a tiled stove, and the wine served in green-stemmed glasses.

The museum across the square from the cathedral holds what came off it: original statues taken down and replaced with copies, the stained glass, the architects' drawings on parchment, and rooms of Upper Rhine painting and sculpture from the fourteenth to the sixteenth century. It is arranged through a set of old houses and courtyards, so you go up and down stairs constantly and lose your bearings, which suits it.

In the afternoon walk west into the Petite France, where the tanners, millers and fishermen worked because the river split into channels here and the water did the work. The houses are timber-framed with steep roofs full of open lofts where the hides were dried, and the whole quarter is now four canals, several locks and a great many restaurant terraces.

At its western end the river is crossed by three medieval bridges under four squat towers, which is what is left of the fourteenth century defences, and then by a long low dam Vauban put across the whole river in the 1680s so the city could flood the southern approaches if it was attacked. Its roof is a terrace, free and open, and it gives you the towers, the canals and the cathedral lined up behind them.

Two kilometres east of the cathedral, in an area of water and parkland, sits the working half of Strasbourg. The European Parliament meets here twelve times a year, moving its entire operation from Brussels for a week at a time, and the building was designed to look unfinished on purpose, as a statement that Europe is not a completed project. During a plenary week you can watch from the public gallery; the rest of the time the visitor centre next door does the job.

The Council of Europe and the human rights court are within a few hundred metres, the second of them a Richard Rogers building from 1995 with two drums for its courtrooms that looks like a ship. None of this requires more than a morning and all of it is free.

Across the road is the oldest park in the city, laid out in 1804 and given to Josephine Bonaparte's taste. It has a lake, a boating pond and, more to the point, storks: the bird is the emblem of Alsace, was nearly extinct here by the 1970s, and was brought back through a breeding programme that started in this park. There are usually a dozen nests visible. Take the tram back into town in the afternoon for the museum of Alsatian daily life, which fills three linked houses with the furniture, costume, kitchen equipment and religious objects of ordinary people from the thirteenth to the nineteenth century, Christian and Jewish alike.

Half an hour south by train and the scale drops. Colmar has around seventy thousand people, an old centre you can cross in fifteen minutes, and a completeness that comes from not having been bombed: the town was taken almost intact in 1945 after the fighting went around it.

Start with lunch in the covered market, a brick and iron hall from 1865 that spent decades as something else and went back to being a market in 2010. Twenty or so stalls sell cheese, charcuterie, bread and vegetables, several of them serve at the counter, and it sits directly on the water at the top of the canal quarter.

Then walk. The Pfister house on the corner of the main street dates from 1537 and has a wooden gallery, a corner oriel and a painted facade, and it is the building every photograph of Colmar starts with. From there the lanes drop to the Lauch, where the old tanners' and market gardeners' quarter has been renamed Little Venice by somebody in tourism and is nonetheless very pretty at dusk, with flat-bottomed boats and a footbridge every fifty metres.

The Unterlinden occupies a thirteenth century Dominican convent with a new wing added by the architects of Tate Modern in 2015, and one object in it justifies the whole trip. The Isenheim altarpiece was painted between 1512 and 1516 by Matthias Grunewald for a monastery hospital that treated ergotism, a disease that rotted the limbs of the poor who ate infected rye. The Christ on the cross is covered in the same sores the patients had, his hands are twisted, and the wood is greenish. It was designed to open in stages, so the sick saw first the agony, then the resurrection. Nothing else in northern painting is like it, and it is displayed in the convent chapel with room to stand back.

Five minutes away, in a former Dominican church, hangs Martin Schongauer's Madonna of 1473: red roses, gold ground, and a tenderness that is the exact opposite of what you have just seen. The two paintings are half a kilometre and forty years apart and seeing them in one morning is the point of coming to Colmar.

In the afternoon, the house where Auguste Bartholdi was born in 1834. He designed the Statue of Liberty, and the museum keeps his working models and drawings, including early plasters where she holds the torch differently. There is a full-size ear somewhere in the collection, which puts the scale of the thing into perspective better than any photograph.

Colmar to Basel takes about fifty minutes with a change at Mulhouse, and the border arrives without ceremony. Basel's main station has a Swiss half and a French half under one roof, which is the first sign that this city treats three countries as one place: people live in Germany, work in Switzerland and shop in France, and the trams cross the frontier.

The cathedral on the terrace above the Rhine was begun in 1019, knocked down by an earthquake in 1356 and finished around 1500 in red sandstone with a coloured tile roof. It has been Protestant since the Reformation, which stripped it out, and the man buried under a slab in the north aisle is Erasmus, who came to Basel to have his books printed, died here in 1536 and was given a tomb in a cathedral that had just changed religion. Behind it, the Pfalz terrace hangs over the river and is where the city sits in the evening.

Then go along the bank to the museum of Jean Tinguely, who spent his life building machines out of scrap that do nothing at all except move, rattle, draw and occasionally destroy themselves. You press a pedal and a two-metre contraption of bicycle wheels and animal skulls comes to life. The building is by Mario Botta and the collection is the least reverent art museum in Switzerland.

In 1661 the city of Basel bought a private cabinet of paintings and curiosities to stop it being sold abroad, and put it on public display. That makes the city's art collection the oldest municipal one anywhere, and it has kept buying. Holbein, who worked in this city, is here in depth. So are the Rhenish primitives, a floor of Impressionists, a room of Picasso the city bought after a public vote in 1967, and a modern annexe across the road on the same ticket.

The Picasso vote is worth knowing about. Two paintings on loan were about to be sold; the city offered half the money, the citizens approved the rest in a referendum, and Picasso was so pleased that he gave the museum four more works. There is a plaque about it, and it explains the town.

In the afternoon take the tram twenty minutes out to Riehen, where a long low building by Renzo Piano sits in a park with a lily pond outside it. The Beyeler collection was assembled by a dealer who kept the best of what passed through his hands: Rothko, Bacon, Giacometti, a wall of Monet water lilies positioned so the pond outside continues the painting. It is arguably the finest small museum in Europe and it is set up so you can see it in two hours.

An hour south and the Alps appear on the horizon. Bern is built on a tongue of rock wrapped almost all the way round by the Aare, entirely in the same green sandstone, with six kilometres of arcades along its main streets so that the whole capital can be crossed under cover. There are more than a hundred fountains, eleven of them Renaissance figures from the 1540s standing in the middle of the road, including one of an ogre eating a child that nobody has satisfactorily explained.

The Zytglogge stands where the western gate used to be. It has stood since the early thirteenth century and served as a prison for women convicted of sleeping with priests, and since 1530 it has carried an astronomical calendar clock whose figures, a jester, a rooster, a procession of bears and a golden man with a hammer, perform four minutes before every hour. The crowd gathers on the eastern side; the mechanism, which you can visit on a tour, is more interesting than the show.

Three doors down the arcade, on the second floor of an ordinary house, is the flat where Albert Einstein lived while he worked as a clerk at the patent office. In 1905 he published four papers from this address that between them established special relativity, the photoelectric effect and the reality of atoms. The flat is furnished as it was and takes twenty minutes, and standing in a small dark sitting room on a shopping street is a strange way to meet the year physics changed.

Renzo Piano's second building on this trip sits in a field on the eastern edge of Bern and looks like three waves of a hill that have been lifted up. Inside is the largest collection of Paul Klee anywhere: four thousand works, of which a rotating selection is shown, because most of them are on paper and cannot take the light. Klee was born near Bern, taught at the Bauhaus, was driven out of Germany as a degenerate artist in 1933 and came back to Switzerland for his last seven years. The last rooms, painted when he was ill and working in thick black lines, are the ones that stay with you.

Walk back into town along the river. Below the bridge on the eastern side, on the bank itself, is the bear park: the city has kept bears since the sixteenth century because of its name and its coat of arms, and the modern enclosure runs down the slope to the water so they can swim. It is still slightly odd and the bears seem content.

Above it, up a path through the trees, is a former cemetery converted in 1913 into a rose garden, with two hundred beds of roses and irises and the best view in the city: the whole sandstone peninsula wrapped in its river, spires and towers and one long ridge of roofs. Get there in the late afternoon.

The Bundeshaus was finished in 1902 and sits on a terrace at the southern edge of the old town with the Alps in front of it, which is a piece of staging nobody in Bern will admit to. The dome separates the two chambers of the Swiss parliament, the guided tours are free, and outside on the square twenty six jets of water come out of the paving, one for each canton, and children run through them all summer.

Walk round to the terrace behind the building before you go. The view runs across the river to the Bernese Alps, and it is the only place in the country where the government, the mountains and a man selling sausages are in the same photograph.