Ten days in Sweden: south down the country until Denmark is across the water

Gamla Stan is close to the station and it makes sense of everything else. The old town is a small island of tall ochre and blood red houses on lanes barely wide enough for two people, and the first written mention of the town dates from 1252. It is also the most visited square kilometre in Sweden, so the late afternoon is the time to be in it.

Stortorget is the middle, and the gabled houses on the north side are the backdrop to the worst two days in Swedish history. In November 1520 the Danish king Christian II, having just been crowned here, had about a hundred Swedish nobles and clergy executed on this square. The outrage put Gustav Vasa on the throne three years later and took Sweden out of the union with Denmark, which is a long way of saying that the country began with a massacre on a square where people now queue for ice cream.

The Nobel Prize Museum is on the same square in the old stock exchange. It is one floor, small, and better than it sounds: objects donated by laureates travel past on an overhead conveyor, and the café serves the ice cream from the banquet. Ninety minutes, then get lost in the lanes deliberately.

The palace went up between 1697 and 1754 after fire took the medieval castle, and it is one of the largest still in use by a head of state anywhere. The royal family live out at Drottningholm, which means the state apartments, the treasury with the regalia and the museum of the burnt castle in the cellars are all open on one ticket. Two hours is honest, and the guard changes in the outer courtyard around midday.

Walk west to Riddarholmen, the small island of noble palaces almost nobody crosses to. The church there is the oldest building in Stockholm, a late fourteenth century abbey with an iron openwork spire, and it stopped being a parish church long ago. It is now a mausoleum: most Swedish monarchs from Gustavus Adolphus to Gustav V lie in it in black and green marble, with the arms of the Order of the Seraphim hung round the walls.

In the afternoon cross the water to the City Hall, the dark red brick building with the square tower that appears in every photograph of the city taken from the south. The Nobel banquet is held in the Blue Hall each December and the dancing afterwards upstairs in the Golden Hall, which is lined with eighteen million gilded mosaic tiles. Inside is by guided tour only and the tower is a separate ticket.

Djurgården is a royal hunting park that became the museum island. Take the ferry from Slussen rather than the bus: it runs every quarter of an hour, costs the same as any other journey and puts the whole waterfront in front of you on the way over.

Go to the Vasa at opening and go first. The ship was the flagship of the Swedish navy, ordered by a king in a hurry, and it sailed thirteen hundred metres on its maiden voyage in August 1628 before a gust laid it over and water came in through the open gunports. It sat in the harbour mud for three hundred and thirty three years, which is exactly what saved it, because the Baltic is too brackish for the shipworm that eats wooden wrecks everywhere else. It came up in 1961 and almost all the timber in it is original. Nothing prepares you for the scale of the thing in the room.

Then Skansen, twenty minutes uphill, which opened in 1891 and invented the open air museum. A hundred and fifty buildings have been taken apart in the Swedish countryside and put back together here, a Dalarna farmstead, a whole town block of workshops, a wooden church people still marry in, with staff doing the work that went with them. There is also a zoo of Nordic animals, so the wolves, lynx and wolverines you will never see in the wild are here. It is big and hilly; choose half of it.

The ABBA museum is by the funfair at the bottom of the hill and is entirely unembarrassed about what it is.

Twenty minutes south on the green line is the strangest thing on this list and one of the best. The Woodland Cemetery was won in competition in 1915 by two architects who were both thirty at the time, and it is on the World Heritage list not for a building but for what they did with a pine forest and a slope. The path from the gate runs along an open meadow towards a low hill with a stand of trees and a plain granite cross beside the way, and the composition is designed to make the walk itself the point. The graves are in the woods, unfenced, under the pines. Greta Garbo is here.

Go early, allow ninety minutes, and treat it as a working cemetery, because it is one.

Come back up the line to Södermalm for the afternoon. The southern island is a granite ridge with the city built up its sides, which is why it has the views. Monteliusvägen is a narrow path along the western cliff top with benches looking straight across at the City Hall and the old town. Around it are the second hand shops of SoFo, a great deal of coffee and the ordinary business of a neighbourhood people live in.

Pack tonight. The Gothenburg train is three hours and worth catching in the morning.

Three hours across the middle of the country, past Lake Vättern, and out to the west coast. Drop the bags and walk to Haga.

Haga was the first suburb outside the city walls, laid out in the 1640s for workers, and by the 1970s it was slums scheduled for demolition. Enough people objected that most of it survived. What stands now is a run of nineteenth century houses built in a very particular local way: a ground floor of stone, because the fire regulations demanded it, and two floors of painted timber above.

It is also where Gothenburg conducts a running argument about cinnamon buns. Several places sell a hagabulle the size of a dinner plate, which is unquestionably a tourist product and also genuinely good, and one is enough for two people. Have the coffee with it, because fika is a fixed part of the day here rather than a break from it.

Then climb the hill behind. Skansen Kronan is a squat seventeenth century fort with a crown on its roof, built to defend the city from the Danes, who never attacked, so it has never fired a shot in anger. The grass around it is where the city sits in the evening looking out over the roofs.

Start on the water, because the canals are why the city is shaped the way it is. Gothenburg was founded in 1621 by a Swedish king who wanted a west coast port the Danes did not control, and he hired Dutch engineers, who did what Dutch engineers do and dug a grid on flat ground. The tourist boats leave from Kungsportsplatsen, run about fifty minutes, and take you along the moat, through the remaining canals and out into the harbour. There is a bridge with roughly twenty centimetres of clearance where everybody is told to lie down and everybody does.

The City Museum is two minutes from the landing, in the building the Swedish East India Company built for itself in the 1750s. That company made a fortune bringing tea, porcelain and silk back from Canton, and its warehouse now holds the story of the region from prehistory to the shipyards. It also has the Äskekärr ship, the only Viking ship ever found in Sweden, a fragmentary thing but the real article.

In the afternoon walk along the quay to Maritiman, which calls itself the world's largest floating ship museum and is nineteen vessels tied together with gangways between them. You go through a destroyer, a lightship, tugs, a monitor and, best of all, a submarine where the engine room is the width of your shoulders and the crew slept between the torpedoes. Two hours, and not for the claustrophobic.

Slottsskogen is a hundred and thirty hectares of parkland ten minutes out by tram, laid out in the 1870s in the English manner, and it is free in every sense: an observatory, a children's farm, a natural history museum that charges nothing, and a small enclosure of Nordic animals where you can look at a moose from about ten metres, which is nearer than anybody sensible wants to be to one in the wild. In late spring the azalea valley comes out and half the city walks through it.

Cross the road to the Botanical Garden, which is on a different scale: a hundred and seventy five hectares running up the slope into a nature reserve, founded in 1923, with a rock garden built into an old quarry, an arboretum, glasshouses of orchids and one of the last surviving Easter Island toromiro trees, a species extinct in the wild.

In the evening, Liseberg. It is the largest amusement park in the Nordic countries and the most visited attraction in Sweden, it opened in 1923, and it is built up a wooded hillside with the rides threaded between pines. Balder, the wooden coaster, is regularly voted among the best in the world by people who take that seriously. Go after seven when the light drops and the lanterns come on, and buy the ride wristband rather than singles.

Two and a half hours through Skåne, which is flat, agricultural and looks more like Denmark than Sweden, because for most of its history it was.

Stortorget in Malmö was laid out in the 1530s by clearing a block of the medieval town, and it was one of the largest squares in northern Europe when it was made. The town hall on the east side dates from 1546 behind a Dutch Renaissance front added three hundred years later, and the stepped gable house in the corner belonged to Jörgen Kock, a German immigrant who became mayor and made his fortune by the straightforward method of running the royal mint. In the middle is a bronze of Karl X Gustav, the Swedish king who took the town off the Danes, on a horse, looking pleased.

Lilla Torg, through an archway to the south west, is the opposite in every way: small, irregular, laid out in the 1590s as the market overflow, ringed with half timbered merchant houses and now given over entirely to outdoor tables.

Keep going west into Gamla Väster, where the medieval street plan survives: low painted houses, cobbles, galleries and design shops, almost no traffic and almost nobody about.

Malmöhus is the oldest surviving Renaissance castle in Scandinavia, built by a Danish king in the 1530s over an earlier fortress, and it has had an undignified career: royal residence, then fortress, then for eighty years a prison. It is now four museums at once, art, natural history, city history and an aquarium, all on one ticket, which makes for an odd ninety minutes in which you go from Nordic modernist painting to a tank of tropical fish without leaving the building.

From there, walk west along the canal and the old fortification line, through the park and out to the sea. It is about twenty minutes to Ribersborg, two kilometres of made beach with shallow water, grass behind it and the whole of Malmö on it the moment the temperature reaches twenty. On a pier at the far end is a bathing house from 1902 with separate men's and women's sections, wood fired saunas, no swimming costumes and steps down into the Öresund. It is open all year and in January people do it anyway.

Carry on north into Västra Hamnen, built on the site of the Kockums shipyard, where the Turning Torso stands: nine cubes of five storeys each, rotated ninety degrees from bottom to top, a hundred and ninety metres high and finished in 2005. It is flats, so there is nothing to go up for. The promenade below it is where the city drinks with the bridge to Denmark on the horizon.

One last thing, and it is the best in the city. St Peter's, two streets from the main square, is a fourteenth century brick Gothic hall church on the Lübeck pattern, which is what northern German merchants built wherever they traded, and it is Malmö's oldest building. The interior was whitewashed at the Reformation, but the small merchants' chapel off the north aisle was walled up instead and kept its medieval painting complete: vines, saints and grotesques over every surface. Fifteen minutes.

Then the station, which is ten minutes north. Trains to Copenhagen leave several times an hour, take thirty five minutes and reach the airport in twenty, running over the Øresund bridge and then down into a tunnel under the western half of the channel.

Take a passport or a national identity card. The checks on the Swedish side are real, and they are checked before boarding rather than on arrival, so leaving them at the hotel is a problem rather than an inconvenience.