Taste extra virgin oils made from Picual olives at an almazara, an olive mill, in the province

Visit Jaén Cathedral, the Renaissance masterpiece by Andrés de Vandelvira begun in 1570, which holds the Veil of Veronica

Visit the Arab Baths under the Villardompardo Palace, among the largest Islamic bathhouses preserved in Spain

Drive up to the Castillo de Santa Catalina at 820m, above the city and the olive groves

Jaén is the largest producer of olive oil in the world — one province, outproducing whole countries — and the trees cover the hills in every direction to the horizon, a monoculture that has shaped everything about how people here live and vote. The dominant variety is Picual, high in polyphenols, stable and distinctly bitter and peppery. Tasting it properly at a mill, out of a blue glass so you cannot judge by colour, is a different thing from buying it in a supermarket.

Jaén: Vandelvira's Cathedral and the Olive Oil Capital

Walk the excavated streets inside the walls and work out that this was the whole town, not a garrison

Work out the dates: besieged 1340, starved into surrender 1341, front line until 1492, then gradually abandoned because it was no longer needed

Go into the abbey church and look down through the glass floor at the graves under it

Climb the keep for the view towards Granada, which is what this place was watching

Work out why the town moved downhill and what that tells you about what walls were for

Alfonso XI besieged Alcalá la Real in 1340 and it capitulated to hunger in 1341, becoming from then until 1492 one of Castile's principal frontier posts facing the Nasrid kingdom of Granada — and Castile paid for it generously, adding works such as the Torre Nueva. The whole medieval town lived inside the walls; once Granada fell, the fortress was let go and the population drifted down to the suburbs on the plain, so what you walk through now is the excavated street plan of a town that emptied itself. The church inside has a glass floor over the burials beneath it.

Alcalá la Real: A Frontier Fortress for Two Centuries

Find the Jabalquinto palace and look at the carved stone spikes covering its facade

Go into the cathedral and find Vandelvira's work inside an earlier building

Find the old university of 1538 and the classroom where Machado taught French

Work out why the town stopped in the 17th century and what that preserved

Baeza was listed by UNESCO together with Úbeda on 3 July 2003, and the two towns are usually done together, but Baeza is the quieter and in some ways the stranger of the pair: a university town from 1538, with a cathedral rebuilt by Andrés de Vandelvira, the Jabalquinto palace in Isabelline Gothic with a facade of carved stone spikes, and a compact grid of Renaissance buildings that stopped growing in the 17th century and never resumed. Antonio Machado taught French here from 1912 to 1919 after his wife died and wrote much of Campos de Castilla in the town.

Baeza: Renaissance in a Small Town

Stand in the Plaza Vázquez de Molina and work out that this is a Spanish hill town built as an Italian one

Go into the Sacra Capilla del Salvador, ordered in 1536 as Cobos's own funerary chapel — designed by Diego de Siloé on neo-Pythagorean proportions and built by Vandelvira from 1540

Find the Hospital de Santiago, begun in 1562 and sometimes called the Andalusian Escorial

Work out who paid for it: the emperor's secretary, spending court money on his home town

Francisco de los Cobos was secretary to Charles V, and he brought Italian taste home to a hill town in Jaén. Working through the architect Andrés de Vandelvira and his followers, Úbeda filled up with palaces and chapels as though it were a small Italian court, and neighbouring Baeza did much the same. Diego de los Cobos drew up the founding statutes of the Hospital de Santiago in 1562 and building started. The two towns were listed together by UNESCO on 3 July 2003 for the quality and preservation of their Renaissance architecture.

Úbeda: A Secretary's Renaissance

Go to the Fuente del Rey and count the spouts — there are well over a hundred

Go into the Sagrario chapel of the Asunción and work out why Cordoban Baroque is spoken of separately

Walk the Barrio de la Villa, the whitewashed medieval quarter on the edge of the cliff

Find out where the money came from: silk, taffeta and velvet sold as far as the Indies in the 18th century

Work out why it is called the city of water once you have found how many springs rise around it

Priego is called the city of water for the springs that rise all round it, and the jewel of Cordoban Baroque for what it built with the money. In the 18th century, after the Moriscos were expelled from the Puerta Granada quarter, it became one of the most important silk centres in Spain, selling taffeta and velvet to Seville, Málaga, Navarre, Portugal, France and the Indies — and that money paid for the churches. The Fuente del Rey has well over a hundred spouts. It sits in the middle of the Sierras Subbéticas, on the Route of the Caliphate.

Priego de Córdoba: Baroque and Water

Climb to the Cueva de los Murciélagos and find the three zones — the lived-in vestibule, the painted sanctuary and the burials at the back

Work out what the 1962 and 1969 digs established: that the Andalusian Neolithic began a thousand years earlier than anyone thought

Go into the Muslim castle built directly onto the crag above the houses

Ride or walk the Vía Verde de la Subbética through the tunnels and over the viaducts below the village

Eat the local sheep's cheese, which is what Zuheros sells beyond its own valley

Zuheros is a white village on a crag in the Sierras Subbéticas with a Muslim castle built into the rock above it, and above that is the Cueva de los Murciélagos. Excavations in 1962 and 1969 pushed the start of the Andalusian Neolithic back by a thousand years. The cave divides into three: a lit vestibule where people actually lived, a sanctuary with Neolithic and Chalcolithic paintings, and a burial zone at the back. Homo sapiens also left engravings and paintings of horses and deer here from around 18,000 years ago. The Vía Verde de la Subbética, on an abandoned railway, runs along the valley below through tunnels and over viaducts. The village also makes a sheep's cheese that it is known for across the province.

Zuheros and the Cave of the Bats

Get to the Mirador de las Peñas and work out what National Geographic was looking at in 2015

Go into the Iglesia de la Encarnación, a round neoclassical church built on the model of the Pantheon

Climb to the Nasrid castle on the rock and find the church the Catholic Monarchs put inside it

Work out what the name means and whether the village lives up to it in January

Eat the local cheese and the morcilla, which is what the inland Granada kitchen runs on

In October 2015 National Geographic put Montefrío on a list of the ten best village views in the world, and the reason is a single composition: a round neoclassical church in the middle of the town, modelled on the Pantheon, with a Nasrid castle on a bare rock directly above it, and white houses filling everything between. The Catholic Monarchs built a church inside the castle after taking it, as they generally did. The name means exactly what it looks like — cold mountain.

Montefrío: Ten Best Views in the World

Take the Teleférico del Teide cable car to 3555m and photograph the cloud layer below — on a clear night the neighbouring islands of La Palma, La Gomera and El Hierro are visible from the upper platform

Watch sunrise from the Roques de García volcanic formations on the Las Cañadas caldera floor — the lava geometry in pre-dawn light is one of the most extraordinary landscapes in the Atlantic

Visit the Observatorio del Teide on a public guided tour — the IAC observatory on Teide's flank is one of the best-equipped professional observatories accessible to the public in Europe

Drive the TF-21 road through the caldera to the Boca Tauce viewpoint at midnight and photograph the Milky Way above the Teide silhouette

Spain's highest mountain rises from the Atlantic above Tenerife in a UNESCO-protected caldera — El Teide holds Class 1 dark sky status, where professional astronomical observatories share the volcano's shoulder with a public cable car reaching 3555m

El Teide: A Cable Car to 3,555m and Stars Over the Caldera

Take a windsurfing or kitesurfing lesson on Playa de Valdevaqueros or Playa de los Lances, where the Levante and Poniente winds blow for over 300 days most years

Take a whale-watching boat trip in the Strait of Gibraltar to see sperm whales, pilot whales, dolphins and orcas

Visit the Castillo de Guzmán el Bueno in the old town, built around 960 with a 13th-century tower

Stand at the Punta de Tarifa, the southernmost point of continental Europe, and look across the Strait to Morocco

Tarifa sits at the narrowest point of the Strait of Gibraltar, fourteen kilometres from Africa, and the Atlantic and the Mediterranean squeeze past each other through the gap — which is why the Levante and Poniente blow here over three hundred days a year and why the town is the kitesurfing capital of Europe. The same channel funnels marine life: sperm whales, pilot whales, dolphins and orcas all pass, and it is one of the best places in Europe to see them from a small boat.