Four days in Asunción: the capital nobody stops in, where the language is Guaraní

Start at the Panteón Nacional de los Héroes, which sits on a corner in the middle of downtown and is the closest thing Paraguay has to a national shrine. It was modelled on the Invalides in Paris, begun in the 1860s, interrupted by the war that nearly destroyed the country, and not finished until the 1930s. Inside are the tomb of the unknown soldier and the remains of the marshals and presidents, with a guard that changes on alternate days and takes it entirely seriously.

Two streets away is the Casa de la Independencia, a low white house of 1772 with a tiled floor and a courtyard, where in May 1811 a small group of men spent a night arranging the removal of the Spanish governor. It worked, and unlike most of the risings of those years it was never reversed. The rooms are furnished as they would have been and it takes half an hour.

Walk down towards the river in the last of the light for the Palacio de López, the presidential palace built in the 1860s facing the water. The interior is closed, which does not matter: the building is meant to be looked at from outside and it is floodlit after dark, which is when the whole thing works.

The Centro Cultural El Cabildo occupies the building that housed the Paraguayan congress until 2003, above the river at the bottom of the old centre. It is now several floors of exhibition space covering indigenous work, colonial religious carving, popular art and temporary shows, and it is one of the few places in town that opens at the weekend.

Next to it, the Manzana de la Rivera is an entire block of interconnected colonial houses turned into a cultural complex, with a small museum of the city, changing exhibitions, a library and an outdoor café looking straight across at the presidential palace. It is the pleasantest hour downtown.

In the afternoon take a taxi east to the Museo del Barro. It is genuinely the best museum in the country and the only one that would stand comparison anywhere: pre-Columbian ceramics, indigenous featherwork and masks, the carved santos of colonial Paraguay, and contemporary Paraguayan painting, all shown together on the argument that they belong to one continuous tradition rather than three separate ones. Allow two hours and check the opening days, which are limited.

Areguá is about half an hour east on the shore of Lake Ypacaraí, and it is where Asunción goes at the weekend. It is a small town of colonial houses on a hill, known for strawberries in winter and for ceramics all year: the road up from the lake is lined with stalls selling pots, figures and garden ware fired locally, and prices are a fraction of the capital's.

The reason to come, though, is Cerro Koi on the edge of town. The hill is made of sandstone that has cracked into hexagonal columns, a formation geologists say occurs in only a handful of places on earth, and you can walk up and put your hand on it. It is a protected area, it takes an hour, and there is almost never anybody else there.

Then work back down through the town. The Casa Amarilla is part gallery, part antique shop and part guesthouse in a building that belonged to the López family estate, and it is far stranger and better inside than the front suggests. The church of La Candelaria at the top of the hill looks over the lake, which is beautiful from a distance and too polluted to swim in.

The Estación Central del Ferrocarril is six blocks from the pantheon and is worth the visit for the building rather than the exhibits. Paraguay laid one of the first railways on the continent in the 1860s, ran it with British-built steam locomotives, and kept them running far longer than anywhere else. The station is now a museum with a wooden ticket hall, the original benches, and carriages and engines standing at the platform.

Then Mercado 4, a few blocks southeast, which is where the city actually buys things: several streets of stalls selling produce, herbs for tereré, clothing, tools, electronics of uncertain provenance and cooked food. It is loud, crowded and completely unpolished. Take a little cash and nothing else, and go in the morning.

The airport is out at Luque, thirty minutes or so northeast, and the road passes the big park where the city runs in the early morning. Allow more time than the distance suggests at rush hour.