Five days in Stockholm: they raised a sunken warship whole and built a museum round it

Start in the old town because it is close to the station and because it makes sense of the rest. Gamla Stan is a small island of tall ochre and blood red houses on lanes barely wide enough for two people, and the first written record of the town dates from 1252. It is also the most heavily visited square kilometre in Sweden, so aim for the early evening if you arrive in the afternoon and the crowds will already be thinning.

Stortorget is the middle of it, and the pretty gabled houses on the north side are the backdrop to the worst afternoon 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 in this square over two days. 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 Sweden as a country began with a massacre on a square where people now eat ice cream.

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

The palace was built between 1697 and 1754 after fire destroyed the medieval castle, and it is one of the largest still in use by a head of state anywhere. The royal family does not actually live here, having moved out to Drottningholm, which means the state apartments, the treasury with the regalia in it and the museum of the burnt down castle in the cellars are all open on one ticket. Two hours is honest. The guard changes in the outer courtyard around midday.

Walk west across to Riddarholmen, the small island of noble palaces that 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 are in it, in sarcophagi of black and green marble, and the walls are hung with the coats of arms of the Order of the Seraphim.

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

Djurgården is a royal hunting park that became the museum island, and it holds more visitors than the rest of Stockholm put together. Take the ferry from Slussen rather than the bus. It runs every fifteen minutes, costs the same as any other SL journey and puts the whole waterfront in front of you on the way over.

Go to the Vasa first and go at opening. 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 the water came in through the open gunports. It sat in the mud of the harbour for three hundred and thirty three years, which is exactly what preserved it, because the Baltic is too brackish for the shipworm that eats wooden wrecks everywhere else. It was raised in 1961 and almost all of the timber in it is the original. Nothing prepares you for the scale in the room.

Then Skansen, twenty minutes' walk 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 rebuilt here, a farmstead from Dalarna, a whole town block of workshops, a wooden church people still marry in, with staff in period clothes doing the work that went with them. There is also a zoo of Nordic animals, so the wolves, lynx and wolverines you will not see in the wild are here. It is large and hilly, so pick half of it.

The ABBA museum is next to the funfair at the bottom of the hill and is unembarrassed about what it is. You can sing, badly, in a booth, and the costumes are on display with the tax advice that produced them.

Twenty minutes south on the green line is the strangest thing on this list and one of the best. Skogskyrkogården, 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 on it and a plain granite cross beside the way, and the whole composition is designed to make the walk itself the thing. 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. Around All Saints' Day at the start of November the whole forest floor is covered in candles.

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 means 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, and Fjällgatan does the same from the east with the ferries below. Between them are the second hand shops of SoFo, a great many coffee places and the ordinary business of a neighbourhood that people live in.

If the evening is clear, walk out to Skinnarviksberget, the bare rock above Zinkensdamm, which is where Stockholm goes to sit with a bottle and watch the sun take its time.

Two last things, both indoors, both near the station, both smaller than they should be.

The Hallwyl house was built in the 1890s as a private residence for a count and countess who collected on an industrial scale and then left the entire building, contents and inventory to the state on the condition that nothing was moved. Nothing has been. It is a wealthy family's house with the plates still in the cupboards and the countess's own catalogue on the shelf, which is a very different experience from a palace.

Then, if you have the morning, the Public Library twenty minutes north. Asplund finished it in 1928, the same architect as the woodland cemetery, and the main hall is a plain cylinder rising out of a square block with three tiers of open shelving running the whole way round it. It is a working library, so you walk in for nothing, and people are borrowing books while you look up.

The central station is fifteen minutes from either, and the Arlanda trains leave from the same platforms as everything else.