Santa Cruz de Tenerife is a working port city of around two hundred thousand people that happens to have a Calatrava concert hall, the best shopping in the archipelago and a carnival that shuts the place down for a fortnight — and most package visitors never set foot in it. It is not a resort. Nobody sunbathes on the seafront, the hotels are business hotels, and the streets are full of people going to work. That is precisely what makes it worth a day.

The city sits on the flat, dry south-eastern side of the island where the mountains run down to a deep natural harbour. It has been the islands' great Atlantic port since the eighteenth century, and it still is: cruise ships, container traffic, the inter-island ferries and the oil bunkering trade all come through here. Shipping money paid for the plazas, the theatres and the confident nineteenth- century facades along the ramblas.

For a visitor the practical shape is easy. Everything worth seeing lies within a rough triangle between the port, Plaza de España and the market, and you can walk the lot in a morning. The tram to La Laguna leaves from the middle of it, the beach is fifteen minutes up the coast, and the mountains of Anaga start immediately behind the city.

Plaza de España and the lake

Start where everyone starts. Plaza de España is the city's front room, opening straight onto the harbour, and its centrepiece is a shallow artificial saltwater lake laid across the square by the Swiss architects Herzog and de Meuron as part of a remodelling completed in 2008. Children paddle in it, the surface reflects the buildings, and at night it is lit.

Behind it rises the Monumento a los Caídos, the tall cross-shaped monument to the Civil War dead — Franco was military commander of the Canaries in 1936 and launched his part of the coup from here, which the city has never entirely resolved. You can climb it for a view. Around the edges are the Cabildo building, the tourist office and the start of Calle Castillo.

The Auditorio: the white wave

Walk ten minutes west along the seafront and you reach the building that put Santa Cruz on postcards. The Auditorio de Tenerife, designed by Santiago Calatrava and opened in 2003, is a white concrete shell with an enormous curved crest rising over it like a wave caught mid-break, sitting on a platform above the Atlantic.

Photographs do not prepare you for the scale, and the trick of the thing — the great crest touches its base at only two points and otherwise hangs in the air — is best appreciated by walking right underneath it. Guided tours of the interior run on a published schedule, and the resident symphony orchestra and opera season mean you can often simply buy a concert ticket, which is a better way to see a concert hall than a tour is. Next door, the Palmetum is a botanical garden of palms built on top of the city's old rubbish tip — a genuinely remarkable piece of reclamation with a view back over the harbour.

TEA and the surrealist connection

Inland from the market, TEA — Tenerife Espacio de las Artes is the same architects' other building here, a long low slab of concrete threaded with slots of daylight, opened in 2008. It holds the city's contemporary art programme, a photography centre, and the permanent collection of Óscar Domínguez, the Tenerife-born surrealist who worked in Paris alongside Breton and Picasso.

The Domínguez link is not incidental. In 1935 Santa Cruz hosted an international surrealist exhibition and André Breton himself came to the island, which for a provincial Atlantic port in the 1930s was extraordinary. The public library inside TEA is open to anyone, free, and is one of the pleasantest places in the city to sit down with a book when the afternoon gets hot.

The market and the shopping streets

Mercado Nuestra Señora de África

The city market opened in 1944, and it does not look like a market from outside — more like a colonial fort, with a clock tower, arched gateway and whitewashed walls, built to a design by José Enrique Marrero Regalado. Inside are two open courtyards ringed with stalls under the sky.

It is a proper working market rather than a tourist one. Fish downstairs, fruit and vegetables in the courtyards, cheese and cured meat around the edges, flowers by the gate. Go in the morning — Spanish markets wind down by early afternoon and Sunday is the quiet day. Look for the island's own produce: black potatoes, bananas that taste nothing like the supermarket ones, papaya, and the sharp white goat's cheeses covered in our guide to Canarian cheeses. Several stalls will let you taste.

The best cheap lunch in the city is the counter bars in and around the market: a plate of grilled fish or a sandwich, standing up, at a fraction of a seafront price. Islanders eat lunch from about two o'clock, so arrive at one if you want a stool.

Calle Castillo

Calle del Castillo runs inland from Plaza de España and is the main pedestrian shopping street, lined with Spanish high-street chains, department stores, shoe shops and cafés, with a grid of smaller streets either side. It is genuinely the best shopping in the Canary Islands — better than Las Palmas for fashion, and vastly better than anything in the southern resorts, which run to sunglasses and tobacco.

The reason prices feel low is not a duty-free myth. The Canaries sit outside the EU VAT area and charge their own sales tax, IGIC, at a headline rate far below mainland Spanish VAT — our explanation of IGIC sets out what it does and does not cover. Electronics, perfume, alcohol and tobacco show the difference most clearly; clothing rather less. For the wider picture, including what is worth carrying home, see shopping in the islands.

Nelson's arm and the cannon called Tigre

Here is the piece of history the city is quietly proudest of. On the night of 24 July 1797, a British squadron under Rear-Admiral Horatio Nelson attempted to seize Santa Cruz and its treasure. The assault failed comprehensively. The defenders, a mix of militia, regular troops and armed citizens, repelled the landings, and Nelson was hit in the right arm as he stepped ashore. The arm was amputated on board that night — he was writing left-handed within days — and he never took another town by storm.

Santa Cruz is one of very few places that beat Nelson, and it has kept the evidence. The Museo Militar de Almeyda displays a bronze cannon called El Tigre, traditionally credited with firing the shot, along with captured British colours and a courteous correspondence between Nelson and the Spanish commander Gutiérrez, who sent the wounded admiral wine and bread. The city commemorates the date every July. If you visit one small museum here, make it that one — and check its current opening hours, which are those of a military institution rather than a tourist attraction.

Carnival

For two weeks in February — occasionally sliding into early March, since the dates follow Lent — Santa Cruz stops being a business city and becomes something else entirely. The Carnaval de Santa Cruz de Tenerife is one of the largest in the world, routinely mentioned in the same breath as Rio, with which it is formally twinned.

The structure is worth knowing in advance. There are weeks of competitive galas first — the murgas, the comparsas, the rondallas and the election of the Carnival Queen, whose costumes are enormous wheeled constructions requiring their wearers to be pushed. Then the street carnival proper begins with the Cabalgata Anunciadora, runs through several nights of open-air dancing, and ends with the Entierro de la Sardina, the Burial of the Sardine, in which an enormous papier-mâché fish is paraded through the streets by mourners in exaggerated black veils and then burnt.

Everyone dresses up. Not a few people — everyone, including grandmothers and toddlers, and a visitor in ordinary clothes will feel conspicuous. Hotels across the whole north of the island fill months ahead. Our page on Tenerife's carnival covers the dates, the practicalities and how to survive the night buses.

Las Teresitas, and the sand question

The city's beach is not in the city. Playa de las Teresitas lies about seven kilometres north at San Andrés, a long pale-gold curve protected by an offshore breakwater, with palms behind and the Anaga mountains rising directly above it. It is where Santa Cruz goes at the weekend, and it is a proper beach: showers, lifeguards, kiosks, calm shallow water.

The gold sand is not local. Tenerife's natural beaches here are black, and Las Teresitas was engineered in the early 1970s using pale sand widely reported to have been shipped in from the Sahara, then held in place by the breakwater. Nobody hides this. It remains a very good beach, and the fishing village of San Andrés at its northern end has a row of restaurants that do the plainest and best grilled fish in the municipality.

The tram, and getting to La Laguna

The Tenerife tram runs from the middle of Santa Cruz up to San Cristóbal de La Laguna, the old university city and a UNESCO World Heritage site, in a little over half an hour. It is clean, frequent, cheap and the single easiest thing to do here: buy a ticket from the machine on the platform, validate it before boarding, and get off at the cathedral.

The two cities are completely different in character — La Laguna is cool, green, aristocratic and full of students, Santa Cruz is hot, commercial and maritime — and seeing both in a day is the classic north Tenerife outing. Buses from everywhere on the island converge on the Intercambiador interchange beside the tram line; our guide to the TITSA network explains the cards and the useful routes.

Parks, museums and arriving by sea

Arriving by cruise ship

Santa Cruz is a major cruise call, and the berths are unusually convenient: the terminal is a short walk from Plaza de España, so you do not need a shuttle or a taxi to reach the centre. That makes it one of the better ports in the islands for doing things under your own steam — our cruise port page sets out the walk, the options and how far you can realistically get in a day.

If you have a full day and a car, the sensible ambitious choice is not the city at all but the Anaga massif immediately behind it — ancient laurel forest, ridge roads and hamlets, half an hour from the port. The alternative long trip is Teide, which is spectacular and takes most of the day.

Green spaces and a museum for a wet hour

The Parque García Sanabria, up the hill from Calle Castillo, is the city's big public garden — a 1920s park full of subtropical planting, a flower clock, and an unexpected collection of outdoor sculpture left over from a 1973 international exhibition, including work by Henry Moore. It is free, shaded, and full of people at six in the evening.

For an hour indoors, the Museum of Nature and Archaeology holds the islands' most important collection of Guanche material, including mummies. It is the serious answer to the question of who lived here before 1496, and it sits a couple of minutes from the market.

Is it worth a day?

Yes — and specifically a weekday, because Santa Cruz at the weekend is a city with its shutters down. Come for the market in the morning, the shops and TEA in the middle of the day, the Auditorio and the seafront in the late light, and dinner somewhere off Calle Castillo where the menu is in Spanish. Our pick of where to eat on Tenerife includes several in the capital.

If you are staying in the southern resorts, treat it as a long half-day out: motorway most of the way, an hour or so each way, and easily combined with Candelaria and its basilica on the coast road. If you are staying in the north, it is simply your nearest city. Either way it is the part of Tenerife that is not putting on a show for you, which after a week of resort promenades is a considerable relief.