Eight days at Iguazú Falls: Brazil gives you the view, Argentina puts you in it

Curitiba sits at over nine hundred metres and can be genuinely cold, which catches out people who have packed for Brazil. The centre is compact and best walked on the afternoon you land.

Tiradentes Square is where the town started, with the cathedral at the northern end of it, and the streets running north west from there into the Largo da Ordem are the oldest part of the city: cobbles, low colonial houses, and the ruins of a church begun in the eighteenth century and never finished, which stand roofless in a small square and are the best thing in the historic quarter.

Ten minutes east is the Passeio Público, opened in 1886 as the city's first park, with a fence copied from a Paris cemetery gate, a lake with pedal boats and a great many extremely relaxed capybaras.

Eat in the evening at Santa Felicidade, the district settled by Italian families from the Veneto in the 1870s, where the restaurants serve a fixed price parade of polenta, pasta, chicken and grilled meat until you ask them to stop. It is twenty minutes out by taxi and it is what Curitiba does.

Curitiba's modern architecture is worth a day and it is all Oscar Niemeyer's fault. The museum named after him is an enormous white eye on a yellow stalk, added in the early 2000s to a building he had designed decades earlier, and inside it are large galleries of Brazilian design and contemporary art. The building is better than most of what it holds, which Niemeyer would probably have accepted.

Fifteen minutes north, the Ópera de Arame is a theatre made of tubular steel and transparent panels, built in 1992 inside a flooded quarry with a waterfall running past it. It looks like a greenhouse designed by an engineer having a very good day, and you can walk in and around it for nothing when there is no performance.

In the afternoon, the botanical garden, which is the picture on every postcard of this city: a French formal parterre with an art nouveau glasshouse at the end of it, opened in 1991 and modelled on the great Victorian glasshouses. The greenhouse is small; the gardens and the trail through the native woodland behind them are not.

That is a full day of a city that plans things. Tomorrow is a frontier town in the jungle, which does not.

An hour and ten minutes in the air takes you from a cold plateau to a hot, low, green corner where Brazil, Argentina and Paraguay all end. The airport is on the road between the town and the national park, so you will drive past the falls entrance on the way to the hotel.

Do the falls tomorrow morning, not this afternoon: the park is emptiest at nine and you want a whole morning for it.

Instead, go out to the Marco das Três Fronteiras at the end of the day. It is a landmark on a bluff above the confluence of the Iguaçu and the Paraná, and from it you can see the matching obelisks on the Argentine and Paraguayan banks, each painted in its own flag. Cargo barges go past, the sun sets over Paraguay, and there is a reasonable amount of tourist theatre in the form of a show and a restaurant, which you can ignore.

It is a genuinely odd thing to stand in one country and look at two others across a few hundred metres of brown water.

The Brazilian side is one path, about one and a half kilometres long, that runs along the bank opposite the falls and slowly reveals all of them. That sounds modest and it is the better of the two panoramas by a distance: the Argentine side is on top of the water and cannot see itself, while from here the entire two and a half kilometre horseshoe is laid out in front of you, in tiers, with rainbows in the spray most mornings.

The river carries three times the flow of Niagara across two hundred and seventy odd separate drops, and the walk keeps adding more of them until the path descends to a catwalk that runs out into the current below the Devil's Throat. You will be soaked in about ninety seconds. There is a lift at the end and a platform above it for the view back.

Two to three hours covers it comfortably. Go at opening, and the coaches will be arriving as you leave.

In the afternoon, the bird park across the road from the park entrance, which sounds like a filler and is not. It is a serious conservation operation for Atlantic forest species, most of them rescued, and its walk-through aviaries put you inside a net with toucans at eye level and macaws that land on the handrail. An hour and a half.

Across the Friendship Bridge is Ciudad del Este, Paraguay's second city, which exists to sell electronics and perfume to Brazilians and is chaotic in a way that has to be seen once. The border here is famously informal, and thousands cross daily without stopping, but if you are going beyond the shopping streets you should get stamped out of Brazil and into Paraguay properly. Check whether your nationality needs a Paraguayan visa before you go.

The reason to cross is Itaipú. The dam is nearly eight kilometres of concrete across the Paraná, it was for decades the largest power station on earth, and it still supplies the great majority of Paraguay's electricity along with a serious slice of Brazil's. Both countries run tours; the Paraguayan side is cheaper and less polished. The scale is the thing, and the technical tour that goes inside the machine hall needs booking ahead.

Half an hour south of the city is the Salto del Monday, a curtain of water forty five metres high and a hundred and twenty across, dropping into a gorge in the forest. On any other continent it would be the main event. There is a viewing platform at the lip, a path down to the bottom, and on a weekday you may well have it to yourself.

Finish in the shopping streets if that appeals, or come straight back. Crossing the bridge in the late afternoon can take an hour.

The crossing is short and the queue is unpredictable. Twenty kilometres, two immigration posts and a bridge over the Iguaçu, and it can take twenty minutes or the best part of two hours. Go in the middle of the morning, when the buses are quietest, and make sure whatever you are travelling in stops at both posts for stamps.

Puerto Iguazú is a relief after Foz: low, green, quiet, and small enough to walk from one end to the other in twenty minutes. Change enough money for dinner, buy the park ticket for tomorrow online, and set an alarm.

There is one thing in the town itself that is worth going out of your way for, and it is in a back garden. A family have spent decades filling their plot with feeders and native plants, and a few hundred wild hummingbirds now come and go all day. Nothing is caged. You sit on a bench about a metre from birds the size of a thumb and try to photograph something that moves in ways your eye cannot follow. Visits are in small groups and you should book a slot.

Then eat. Argentine beef at Argentine prices, which after Brazil will feel like a bargain.

This is the big day and it needs all of it. Roughly four fifths of the falls are on the Argentine side, and instead of one path opposite them there are three separate circuits that take you out on steel walkways above, beside and underneath the water.

Go to the Devil's Throat first, before the park fills. A little train, or a walk along the track it runs on, takes you to the far end of the park, and then a kilometre of walkway crosses the flat brown river on catwalks to a platform on the exact lip of the largest fall. Half the river goes over the edge in front of you and drops eighty metres into a permanent cloud. The noise makes talking pointless and the spray comes straight back up into your face. It is the single best fifteen minutes on this continent.

Then the upper circuit, which runs along the tops of the smaller falls and lets you stand where the water goes over, and the lower circuit, which descends through the forest to look up at the same falls from below and passes under one of them.

Coatis are everywhere on the circuits and are not tame, whatever they appear to be. Butterflies land on everybody. Toucans turn up in the trees near the station, and if you are on the paths early or late, so occasionally do capuchin monkeys.

Puerto Iguazú's airport is twenty minutes from the town on the road towards the park, and it is small and quick. Direct flights go to Buenos Aires in about two hours, which is the neatest way to leave.

If you are flying out of Brazil instead, allow two hours more than seems reasonable: the bridge queue is worst in the morning, and the taxi drivers who do this run every day will tell you when to leave. Believe them.

One last thing worth doing on the way, if the timing works. The Argentine three borders marker sits above the river junction at the edge of town, and it is the mirror image of the one you stood at on the third evening. Same two rivers, same three obelisks, different country, and a useful reminder that the falls only look like one thing from one side.