Four days in Riga: the Baltic city that got rich first and spent it on facades

Start in the square the Old Town is built around. The House of the Blackheads is the ornate red brick and stone building with the stepped Dutch gable, and it belonged to a guild of unmarried foreign merchants who took St Maurice as their patron, which is where the name comes from. The original went up in the fourteenth century and was extended and decorated for six hundred years. The Germans shelled it in 1941, the Soviets bulldozed what was left in 1948, and the building you are looking at was put up between 1995 and 1999 from photographs and drawings. It is a copy and it does not pretend otherwise, and inside there are ballrooms, a wine cellar and a serious collection about the trade that paid for all of it.

The black basalt block behind the square, which looks like it landed there, is deliberate. That is the occupation museum, and it is tomorrow.

Then take the lift up St Peter's. The church dates to 1209 and is the oldest in the city, and its spire has burnt down, blown down and been shot down repeatedly, most recently in 1941; the current one is a steel reconstruction from the 1970s, which is why there is a lift inside it rather than a staircase. From the platform you get the whole thing: the red roofs of Vecrīga, the Daugava and its bridges, the Zeppelin hangars of the market, the Academy of Sciences and, on a clear day, the sea. Go up an hour before sunset. The lift does not run on Mondays.

Riga Cathedral is the largest medieval church on the eastern Baltic and it was begun within a decade of the city itself, which tells you what the priorities were. The interior is plain to the point of bleakness, because this became a Lutheran church in the Reformation and the Lutherans took everything out. What is left is the organ, built in 1884 with over six and a half thousand pipes, which was the largest in the world when it was new. If there is a lunchtime recital, go to that instead of paying to walk around the empty nave.

North of the cathedral the streets narrow into the oldest surviving fabric in the city. The Three Brothers on Mazā Pils iela are three houses built side by side across three centuries, the right hand one from about 1490 and the oldest dwelling in Riga, the middle one from 1646, the left one narrower and later, and you can read the change in taste straight off the facades. One of them holds the architecture museum, which is free. Round the corner is the Swedish Gate, cut through the city wall in 1698 and the only one of the eight gates still standing.

Give the afternoon to the black building. The Museum of the Occupation of Latvia covers 1940 to 1991: the year of Soviet rule, the deportations of June 1941 when fifteen thousand people were put on trains in a single night, the German occupation and the destruction of Latvian Jewry, the return of the Soviets, the forty five years after. It is long, densely written and genuinely upsetting, and it is the single most important thing to do in Riga. Two hours minimum, and do not schedule anything demanding afterwards.

The Freedom Monument stands where the boulevard crosses the line of the old walls, forty two metres of travertine and copper with a woman at the top holding up three stars for the three historical regions of Latvia. It was paid for by public subscription and unveiled in 1935. What is remarkable is that the Soviets never took it down, though they reinterpreted it shamelessly and made laying flowers at it an arrestable offence, which of course is exactly what people did. There is an honour guard, changed on the hour.

Behind it, Bastion Hill is the earth of the demolished city rampart turned into a park in the 1850s, with the city canal curling through it and a footbridge that has collected several thousand padlocks. There is also a small memorial here to the people shot by Soviet special forces in January 1991, which is not the sort of thing a park like this usually contains.

Then walk north into the Quiet Centre for the reason Riga is on the World Heritage list. Alberta iela is the concentrated version, a single short street of apartment blocks from 1901 to 1908, most of them by Mikhail Eisenstein, whose son Sergei went on to make films. The facades carry screaming faces two floors up, sphinxes, peacocks, women's heads three metres across and blue and gold tilework, and it is worth walking it twice, once looking up and once looking at the doorways. Elizabetes iela and Strēlnieku iela have more of the same. Almost none of it is open to the public: this is a residential district and the thing to do is look.

Finish at the National Museum of Art in the Esplanade park at the end of the street, which is the country's main collection and largely Latvian. The nineteenth century landscape painters and the interwar modernists are the reason to go, and the building itself, a heavy neo-Baroque pile of 1905 with a glass extension buried under the lawn, is worth the ticket on its own.

Riga Central Market is five minutes from the Old Town and it is a working market, not a food court. The five great vaults are surplus German Zeppelin hangars bought in 1920 and re-erected here, each one now a hall with a theme: fish, meat, dairy, vegetables, bread. Outside there are hundreds of stalls under awnings. It is where Riga actually shops, prices are what Latvians pay, and the cafés wedged between the halls do a plate of hot food for a couple of euros. Buy smoked fish, black bread and whatever the mushroom season has produced, and eat it standing up.

The tower behind the market is the Academy of Sciences, finished in 1953, and it is the Stalinist wedding cake skyscraper that every capital in the eastern bloc got one of. Riga's is nicknamed Stalin's birthday cake, it was built partly with collective farm contributions that were not exactly voluntary, and the hammers and sickles are still on the spire. The seventeenth floor balcony is open to the public for a few euros and it gives you the view the Old Town spires cannot, which is the Old Town itself, the river and the whole flat plain around it.

Both are a short walk from the railway station and from the bus station, which makes this a workable last morning whatever time you are leaving.