Using the river as a compass, you can explore many of the city’s best sights.

There is beauty in the way the River Danube neatly splits Budapest in two, like a north-to-south flowing compass. Even the deliberately map-less traveller cannot go wrong using it to explore the city.

Start your river-led adventure at the city's best kept secret, the turbe, or tomb, of Gul Baba, a 16th Century dervish and the northernmost place of pilgrimage for devout Moslems.

The tomb is located on Rozsa domb, or Rose Hill (Gul Baba means “the father of the roses” in Turkish). Close to the western end of Margaret Bridge, find the steep, cobbled Gul Baba Street which winds upward from Leo Frankel Street, then take a sharp left at the top, and look for the new moon symbol on a small, domed roof. In 1526 Gul Baba came to Buda – then still a separate city from Pest, until they were united by the bridges and politics of the 19th Century. He was ahead of the conquering Ottoman armies, and has long outstayed them; he has been locally cherished for generations for the charitable soup kitchen he once ran here. The turbe was restored with donations from the Turkish state in 1998.

There is a good view of the river and valley from here. You can see Margaret Island, Castle Hill, Gellert Hill and the great sprawl of the Pest side of town on the opposite bank.

Not far from the foot of the hill is Bem Square. For a taste of Hungary in Communist times, try the Bambi cafe on the corner (Frankel Leo u. 2; 36-1-212-3171), where the decor, coffee and cake menus, and even some of the clients, have not changed since the 1980s.

Then shut your eyes and imagine the echoes of the revolution of 1956 in the square in front of the cafe. The students converged on Bem Square from both sides of the river, on 23 October that year, to show their solidarity with the people of Poland. (Jozef Bem was a Polish general who helped the Hungarians in their 1848 war of independence.) In 1956, even the Army conscripts in the barracks at Bem Square 3 tore the little red metal stars from their caps and threw them from the upstairs windows, in sympathy with the protesters.

A couple of minutes walk farther on, is Batthany Square. Once the track reopens at the end of this month, and for barely a euro, you can ride the number 19 or 41 tram along the western bank of the Danube all the way to Gellert Square, with an excellent view of the river and the Pest embankment. There are many small cafes and restaurants on the way, but Marvelosa, just past the Chain bridge and the entrance to the tunnel on the right, serves excellent cakes and is unusually friendly for Budapest. There is also a good wine shop next door to replenish your wine cellar.

Once you reach Gellert Square, try the natural spring baths, which are  expensive for a couple of hours, but a bargain for a half or even a whole day. As well as good outdoor and indoor pools, there is an excellent pool for children and plenty of shade under the trees. If you do not get as far as the baths, there is a brasserie to the right of the hotel entrance. Try the Hortobagyi (savoury meat and sour cream) pancakes. The Panorama restaurant on the terrace is more expensive but has an excellent view.

Opposite the entrance to the baths is a walk up Gellert Hill, through a park where you can hear nightingales in early summer.

If you walk over the Szabadsag (Freedom) bridge to the Pest side, the ornate covered market looms on your right (open every day except Sunday),where you can buy strings of garlic or red peppers, or spicy Hungarian salami. Up the iron stairs on a terrace running round the walls are food and drink stalls and shops for traditional, embroidered shirts and tablecloths.

The number 2 tram on the Pest embankment to Vigado Square, where boats leave each hour for 45-minute trips up and down the river. The commentary, in a choice of 14 languages, is informative, but you can shun the headphones and just enjoy the view of the 19th Century houses, universities churches and fine bridges.

Back on the tram, carry on to Parliament where here are guided group tours. The most interesting sight is the Holy Crown of St Stephen, directly beneath Parliament’s dome, but there are also excellent views of the river. Note the brass cigar holders for Parliamentary deputies on the window-ledge, from a more tobacco-friendly age. On the grass next the building is the seated statue of Attila Jozsef, one of the country's great 20th Century poets. His poem, By the Danube, won him a place next to it for eternity.

On the north side of Parliament, near the tram stop and next to a delicatessen, is the hard-to-find patisserie, Szalai (Balassa Balint u 7; 36-1-269-3210)). This cafe has fed generations of protesters with divine treats like the meggyes retes (cherry strudel) and the Svajci kifli, which is a solid croissant-like pastry with icing sugar on top and ground walnuts inside.

The final stop is Margaret Island, in the middle of the Danube. This can be reached by the 4 or 6 tram, both of which stop on the middle, or is a short walk from either bank. There is a swimming pool, a small outdoor zoo, several cafes and hotels, and the island is a favourite sunbathing place for the city inhabitants. It is a great place to take children for a walk and an ice-cream or go for a run.

Nick Thorpe is the author of ‘89, The Unfinished Revolution, a memoir of Eastern Europe. His next book, Danube, will be published by Yale University Press in 2012.