Eleven days on the River Plate: Buenos Aires is the loud one, Montevideo is the one you remember

Rosario is a three hour drive or a short flight from Buenos Aires and it is where this trip starts, on the upstream end of the same water it finishes on.

In February 1812 Manuel Belgrano raised a light blue and white flag on the bank of the Paraná here, on a battery he had built to fire on Spanish ships, and Argentina has been using it ever since. The monument that marks the spot went up in 1957 and is enormous, a stone prow of a ship pointing at the river with a tower you can go up for the view over the whole delta.

The rest of the afternoon belongs to the waterfront. The old port has been turned into a continuous park of several kilometres, and on any warm evening it is full of people running, fishing, drinking mate and doing very little.

A block or two inland, on a corner of Calle Entre Ríos, is the apartment building where Ernesto Guevara was born in 1928. There is nothing to go into and a plaque to read, and it is a strange thing to stand outside.

The contemporary art museum is in a set of grain silos on the riverbank, painted in bands of bright colour and visible from most of the waterfront. Ten floors of a working silo have been turned into small galleries stacked on top of each other, the lift is the size of a phone box, and the top has a bar with a view down the Paraná. The art is Argentine and mostly recent.

In the afternoon, cross town to Parque de la Independencia, which covers a hundred and twenty hectares and contains a boating lake, a rose garden, a racecourse and two football stadiums.

The fine arts museum sits inside it, and it is a genuine surprise: a provincial collection assembled by a grain merchant who bought well, with Spanish, French and Italian paintings alongside the Argentines, in a building the size of a small palace.

Eat late and near the river. Rosario's other speciality, besides beef, is river fish: surubí and pacú, grilled, which you will not find easily anywhere else on this route.

Four hours by coach down the flattest road in the world, and you are at Retiro in the middle of Buenos Aires. Take a taxi or an app car out of the terminal rather than walking; the neighbourhood immediately behind it is not one to arrive in with luggage.

Give the first afternoon to the Plaza de Mayo, which is where Argentina has had every argument it has ever had. The Casa Rosada is at the eastern end, pink supposedly because a nineteenth century president mixed the red of one party with the white of the other, and it is the balcony Perón spoke from. The Mothers of the Plaza de Mayo have circled the pyramid in the middle every Thursday afternoon since 1977, and they still do.

The Cabildo on the other side is the one colonial building left on the square, and the small museum inside it is the clearest twenty minutes you can spend on what this place was before it was any of this.

Then dinner, at an hour that will feel absurd. Nothing here starts before nine.

The Teatro Colón took nearly twenty years to build and opened in 1908, at the point when Argentina was briefly one of the richest countries on earth and wanted an opera house to prove it. The guided tour runs several times an hour, takes fifty minutes, and gets you into the horseshoe auditorium, which seats about two and a half thousand and has acoustics that singers still describe as among the best anywhere.

Then walk down Avenida de Mayo to the Palacio Barolo, which is the strangest building in the city. The architect, Mario Palanti, laid it out as the Divine Comedy: hell in the basement, purgatory up the shaft of the building, and paradise in the lighthouse at the top, with twenty two floors for the twenty two cantos and a hundred metres of height for the hundred cantos in total. Book the tour, go up, and note the name, because you will see his other tower on the far side of the river next week.

Finish at Pasaje Rivarola, two blocks north, a short street where every building on one side is a mirror image of the one facing it. Almost nobody goes and it is one of the best photographs in Buenos Aires.

Recoleta cemetery is several hectares of marble streets, several thousand vaults, and the families who ran Argentina for a century competing with each other for the last time. Some tombs are chapels, some are art nouveau, several are falling in on themselves with the coffins visible through broken glass. Eva Perón is here, in her family's black granite vault, which is smaller than most of the ones around it and the only one with a queue.

Ten minutes away, the National Fine Arts Museum, which is free and considerably better than that suggests: Goya, El Greco, Rodin, a wall of Degas and a good deal of Argentine painting nobody outside the country knows. Two hours is enough.

In the late afternoon, walk to El Ateneo. It was built as a theatre in 1919, became a cinema, and is now a bookshop that left everything in place: the boxes are reading nooks, the painted ceiling is intact, and the café is on the stage behind the original curtain. Buy something, or at least sit in a box.

Recoleta and Palermo either side of it are where Buenos Aires eats best. The evening should not be planned beyond a booking.

La Boca was where the Italians landed and it built its houses from corrugated iron off the ships and painted them with whatever paint was left in the tins, which is why Caminito is the colour it is. It is now heavily commercial, full of tango dancers working for tips, and worth an hour in the middle of the day. Stay on the marked streets, keep the phone away, and take a car in and out.

Two streets up is La Bombonera, which is the least beautiful and most famous football ground in South America. It is nearly vertical, holds around forty nine thousand, and the museum and stadium tour run daily except match days. This is Boca Juniors, and Maradona's stand is a shrine with the paint worn off it.

Come back east for the end of the day. Puerto Madero is the redeveloped dock, all glass towers and converted brick warehouses, and the Calatrava footbridge across the middle basin swings open for boats and is best photographed with the sun behind it.

Behind the docks is the ecological reserve, three hundred and fifty hectares of scrub that grew up on rubble dumped in the river, and it is the only quiet place in central Buenos Aires.

The ferry leaves from Puerto Madero and takes a little over an hour on the fast boat. Allow an extra ninety minutes at the terminal: Argentine and Uruguayan immigration sit at adjacent desks in the departure hall, you are stamped out of one country and into the other in the same queue, and then you are on a boat.

The town was set up by Portugal in 1680, directly across the water from Spanish Buenos Aires, entirely so that goods could be smuggled into the Spanish empire without paying for the privilege. Spain took it, Portugal took it back, and this went on for over a century, which is why the old quarter has Portuguese and Spanish street plans laid over each other: one lot drained the streets down the middle, the other down the sides.

The barrio histórico is small enough to walk in twenty minutes and rewards a whole afternoon. Go through the old gate with its wooden drawbridge, walk the Calle de los Suspiros, which is a cobbled slope of low houses that has appeared on the cover of every book ever written about this country, and climb the lighthouse, which was built in the 1850s on the ruins of a seventeenth century convent.

Then wait. The last day boat takes the crowds back to Buenos Aires in the late afternoon, the town empties, and the sun sets straight down the river at the western bastion with the lights of Argentina coming on across the water.

Two and a half hours by coach along the coast, and Montevideo begins without much warning: low buildings, then taller ones, then the terminal at Tres Cruces with a shopping centre attached.

Plaza Independencia is where to start, because it is the seam of the city. On one side is the last surviving arch of the Spanish citadel, and through it the old town. On the other is the avenue that runs to the modern city. In the middle, under the square, is the marble mausoleum of José Artigas, the man who led the fight against Spain and then against Argentina and Brazil in turn, with a permanent honour guard on the steps down to it.

Above all of it stands the Palacio Salvo, which was the tallest building in South America when it went up in 1928, is Art Deco crossed with something considerably odder, and was designed by Mario Palanti. He is the same architect who built the Barolo in Buenos Aires five years earlier, and the two towers were meant to face each other across a hundred and eighty kilometres of water, one with a lighthouse and one with a beacon, signalling to each other. Nobody has proved that was really the plan, and everybody on both banks tells you it was.

Eat in the old town. It is safer and livelier than its reputation before about ten at night.

The Ciudad Vieja is a grid of about twenty blocks on a small peninsula, faded, occasionally derelict and full of very good early twentieth century architecture that nobody has got round to cleaning. Walk in through the citadel gate and down Sarandí, which is pedestrian all the way.

The Teatro Solís opened in 1856 and is the country's national theatre, and the tour of the auditorium takes forty minutes. Then keep going west to the Mercado del Puerto, an iron and glass market hall shipped from England and put up in 1868, which stopped being a market a long time ago and is now two dozen wood fired grills under one roof. Lunch is beef, cooked on a parrilla the size of a car, with a medio y medio, the local half sparkling wine and half white, to go with it.

In the afternoon, cross to the town hall and go up. The Palacio Municipal has a viewing platform on the roof that most visitors never find, and it gives you the peninsula, the port, the rambla and the hill with the fort on it across the bay in one turn.

The building also holds the city's photography and art history museums, and both are free.

Give the morning to the rambla. It runs the whole southern edge of the city, twenty two kilometres of it, past beaches, headlands and the fishing spots, and walking or cycling a stretch of it is the single most Montevidean thing available. On a Sunday it is packed with people carrying a thermos under one arm and a gourd in the other hand, and nobody is going anywhere in particular.

In the afternoon, the Estadio Centenario. It was built in 1930 for the first football World Cup, in the year Uruguay marked a century of its constitution, and Uruguay won that tournament in this ground in front of ninety odd thousand people. A country of under two million had just beaten Argentina in the final, which explains a great deal about this place.

The football museum is inside it and covers both of the wins that made the country: 1930 here, and 1950 at the Maracanã, where Uruguay beat Brazil in front of the largest crowd ever to watch a football match. The stadium is usually open to walk into when there is no match, and standing on the terrace of an empty 1930 stadium is worth more than the exhibits.

If there is a match, go. Tickets are cheap and the noise is the point.

The Palacio Legislativo is a ten minute drive from the centre and it is worth the last morning. It was finished in 1925 for the centenary of independence and it is built almost entirely of Uruguayan marble, in every colour the country could quarry, by a country that had decided its parliament should look like the most permanent thing in it. The guided tour is short, cheap and runs in English on request.

That confidence is the note to leave on. Uruguay wrote itself a parliament out of coloured stone at a moment when it was the wealthiest and steadiest country in South America, and it has spent the century since being quietly, stubbornly unlike its neighbours.

Carrasco airport is twenty kilometres east along the coast, thirty minutes in normal traffic, and the terminal is one elegant curved shed that takes ten minutes to get through. Give yourself two hours, and spend some of them looking at the building.