We're taking the lid off the world's most underrated travel destination

The hooves of my sturdy Andean pony make a dull metallic ring as they hit the broken shale of the path winding up into the foothills. We are following Elvio, a gaucho with a deeply tanned, weathered face below a jaunty "boina" beret and a walk suggestive of many years in the saddle. In the distance are the serrated peaks of the snow-tipped Andes and above, in a cerulean sky, a lone condor hovering in the air currents, hunting for prey in the pampas grass.

Argentina doesn't feature prominently on most Australians' travel wish lists. In fact, between July 2023 and April this year, only 5000 Australian travellers marked Argentina as their primary destination on their arrival cards. Which seems extraordinary, given the country's wild beauty, its sophisticated cities, exceptional food and wine and the ease of getting there - via Santiago in Chile with flying time just an hour longer to Los Angeles.

With just two weeks, we decide to forgo Argentina's most popular tourist attraction, Iguazu Falls, and divide our time between Buenos Aires and the regions of Patagonia and Mendoza, for a taste of city, mountains and wine country.

Nicknamed the Paris of South America, Buenos Aires, like the French capital, is filled with historic architecture and lively cafes, bars and restaurants. Subbing for arrondissements are 48 barrios, each with a distinct identity.

We begin in one of the best-known, La Boca, with a walking tour. Meaning "the mouth" in Spanish, this riverside barrio was once the city's main harbour, where the majority of immigrants arriving in the 19th century landed. Many worked in the maritime industry and built their tenement homes from leftover shipbuilding materials. Known as "los conventillos" (convents) they were rambling spaces with a small central courtyard and ad hoc shared rooms.

"The housing crisis was so bad that inside of these houses they used to rent beds that were called warm beds, because they were rented by shifts of six or eight hours and once you get inside you can still feel the warmth of the person that was there before you," our guide tells us as we walk through one.

The most photographed street in Buenos Aires is La Boca's El Caminito, where the houses along a pedestrian laneway are painted in primary colours, from hot pink to burnt orange. La Boca is also home to the famous La Bombonera, the Ikea-blue-and-yellow home stadium of one of Argentina's most revered football teams, Boca Juniors.

Buenos Aires' other claim to fame is, of course, as the birthplace of the tango, both a musical genre and a dance, and there are scores of tango dinner shows advertised everywhere. I'm initially sceptical, but a tango academic and dancer who has given us a tour and a fascinating history lesson in the Abasto barrio, where "the father of the tango" Carlos Gardel lived, has recommended El Querandi.

One of the city's oldest and most traditional shows, it turns out to be excellent, giving a history of the tango with some impressive dancing, elaborate costumes and a not-bad-at-all dinner.

In the heart of the city, Plaza del Mayo is the most important square, the place where modern Buenos Aires was established and the scene of many political demonstrations. If you come on a Thursday afternoon you'll witness a poignant ritual enacted by the Grandmothers of Plaza del Mayo, who walk around the square in silent protest. They are women whose children were "disappeared", their fates unknown, taken for political or other reasons by the junta who ruled Argentina between 1976 and 1983. Plaza del Mayo is home to Casa Rosada, "the pink house", the seat of the government. It's from here, on a second-floor balcony, that Juan Peron and his first lady Eva addressed the masses between 1946 and 1952.

While Juan Peron has his own mausoleum at an estate in the far south of the city, Eva Peron is buried in a fairly modest tomb at the extraordinary Recoleta Cemetery. Constructed in 1882, the cemetery is a vast necropolis where upper-class Portenos are interred in elaborate granite and marble crypts, well worth a visit.

On Sunday morning, we wander the Feria de San Telmo in the city's oldest barrio, the bohemian San Telmo. This antique and curio market was established with just a few stalls in the 1970s but now numbers around 250, plus shops in the surrounding cobblestoned streets. It's a vintage lover's dream, with vendors selling collectable items from mid-century furniture to 1970s clothing, glassware, vinyl records and ephemera of all kinds.

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Dine at a parilla, traditional steakhouses like Parilla Don Julio, in the hip and happening Palermo barrio.

Visit one of the world's most beautiful bookshops, El Ateneo Grand Splendid, a former theatre.

It sounds as remote as Timbuktu but in reality Patagonia, the bit of fractured land at the bottom of the world with a border shared by Chile, is less than 3.5 flying hours from the capital.

The region's airport is in El Calafate, a name taken from the bushes of yellow flowers and purple fruit that grow in the area. The town, which has an alpine-village feel, has one main road lined with shops and restaurants and a small weekly artisan's market. We stay at Esplendor El Calafate, a hotel on the hill with views across the milky-blue glacial Lago Argentino to the snow-covered mountains.

Most people come here to see the Perito Moreno glacier. We've been warned the weather can be capricious in Patagonia but the sky is blue and the sun's out, turning the snow drifts on the roadside into mounds of silver-white glitter.

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It's a leisurely drive of around an hour and half to Los Glaciares National Park and the Perito Moreno. Around 30 kilmetres in length, five kilometres wide and 70 metres high, it's one of the world's most accessible glaciers, with a lengthy boardwalk punctuated by viewing points. It's indescribably beautiful, the sheer face translucent blue with vertical slices frequently calving off and crashing into the lake below. We've brought a picnic, purchased at the surprisingly well-equipped grocery store in El Calafate, and drive around to the lake's edge for another viewpoint, crunching across the snow to a table to enjoy our lunch while watching a group of kayakers making their way across the water.

One of the other big drawcards of Patagonia is hiking. The drive to El Chalten, known as the trekking capital of Argentina, takes twice as long as it should because we keep pulling off the road to take photos. It is jaw-droppingly beautiful and dead quiet - we see less than half a dozen other cars on the road. We're headed around Lago Desierto (or Desert Lake) straight towards Fitz Roy, the distinctive jagged peak famous as the logo for Patagonia clothing. To either side are fields full of guanacos, a more elegant-looking relative of the llama, that bound off as soon as we pull over.

Even smaller and more "wild west" in feel than El Calafate, with just 350 permanent residents, the village of El Chalten was built in the late 1980s by the Argentinians to lay claim to this disputed border region. While it usually attracts loads of hikers, being a week out of season, we have the town and our hotel practically to ourselves.

We are not hikers but we do some shorter walks, starting with a trail along the Rio de las Vueltas that carves its way through the canyon to empty into the lake, to Chorillo del Salto waterfall.

In a town where many come and never even see the peak of Fitz Roy, we again hit the lottery with the weather the next day on a hike to Laguna Capri. Apart from an initial steep section it's a relatively easy out-and-back eight kilometres, through woods and across snow-melt streams where tiny spring flowers bloom. We join others at the walk's end, sitting on rocks at a lookout over the blue-green glacial lake and Fitz Roy, everyone communicating in reverent whispers.

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Don't miss the Glaciarium Patagonia Ice Museum, just out of El Calafate. They also have an ice-bar, where you can drink tiny ice glasses of Calafate berry liqueur in minus 15 degrees.

Make sure to try the Patagonian lamb, fed on the rich mountain grass and cooked over flame.

It's Saturday morning and Mendoza is hellishly busy. The city is a grid of confusing one-way streets, and we drive around and around, dodging people and other cars that drift across the road.

We're not staying here in the city but out in the further reaches of the region, close to the Chilean border in the Uspallata Valley near the mighty Aconcagua, the highest peak outside Asia.

Our accommodation is in a glamping dome, on the organic farm of brothers Joaqun and Juan Gonzlez Gaviola. Built on a wooden platform on a drop-off bluff, they have a 360-degree view of the plain, a tributary of the Mendoza River and, in the distance, the spine of the world's longest mountain range. Each has a kitchenette, a wood-fired heater, deck with loungers and a hammock strung beneath the trees.

The Uspallata Valley was once an important trade route for the Inca into Chile and, in more contemporary times, people and goods would travel via Transandine Railway, which opened in 1901. It was abandoned some 80 years later, leaving derelict tracks, railway bridges and old tunnels.

These days the region of Mendoza, with more than 1200 wineries, is most famous for producing the majority of Argentina's wine, including their best-known, Malbec. We call into the stylish Matias Riccitelli with their new-gen wines, including funky "pet nats" (naturally sparkling wines) and other low-intervention drops, and have lunch at nearby Durigatti Family Winemakers in an elegant restaurant among the vines. As the designated non-driver, I accompany my meal with a "mystery flight chosen by the sommelier", with pours so generous I have to leave three or four glasses only half drunk.

The following day is our last, and we take our beginner-friendly ride with Elvio the gaucho, whose own estancia abuts the domes. We don't share a common language but our horses are well-trained and follow his nimble little stead, splashing through streams and across the marshy grasslands, scaring birds into the sky as we go.

There's so much of this naturally beautiful and underrated country we didn't see - the lake district of Bariloche, the salt flats of Salta, the wetlands of Ibera or the majestic Iguazu Falls - but even with this small taste, a return visit to Argentina now tops this Australian's travel wish list.

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Mendoza's wine regions are divided into sub-regions and it's best to concentrate on just one or two in a visit. Make sure to book at winery restaurants.

Bring your swimsuit to visit Cacheuta Spa natural hot water springs, used by the Incas.

Getting there: LATAM flies from Sydney to Buenos Aires from $2276 and Melbourne from $2049 return.

When to go: The best weather in Patagonia is in summer, however this is also high season, and you'll need to book accommodation well in advance. Spring or autumn is cheaper.

Staying there: In shoulder-season October, the well-located Palladio M Gallery in Buenos Aires has deluxe king rooms from $455 a night, all.accor.com; El Splendor in El Calafate has queen rooms from $210, wyndhamhotels.com; Chalten Suites in El Chalten has double rooms with breakfast from $408, chaltensuiteshotel.com/en; Domos Uspallata has queen-sized luxury glamping domes from about $200 including breakfast, domosuspallata.com.ar

Getting around: Aerolineas Argentinas is the national carrier. There is also a budget airline Flybondi. Taxis and Uber in Buenos Aires are extremely well-priced compared to Australia. To travel by Subte (metro) in Buenos Aires, you'll need to get a SUBE travel card and charge it with credit. Europa Car has cars available at the respective airports from about $87 a day in El Calafate and from about $81 a day in Mendoza.

Need to know: Money, specifically the complicated multiple exchange rates, may make your head spin, with the advice being to pay everything by credit card for the best deal. Inflation is rampant in Argentina, with prices for visitors not as cheap as they once were, but still surprisingly affordable, especially getting around.

Explore more: argentina.travel/en

The writer travelled at her own expense.