I stood in front of Picasso's Guernica – much, much bigger than I'd imagined it – the first piece of artwork I laid eyes on in Madrid. If this was to be the beginning of an art-packed long weekend in Spain, it was surely off to a good start. Guernica is flanked on either side by two constant guards, and often too crowded to get a proper look, but we'd arrived early, just after the Museo Nacional Centro de Arte Reina Sofia opened its doors, to take in its twisted, war-anguished power.

It is the 75th anniversary of Picasso's most emblematic work, now the centrepiece of the Reina Sofia's current exhibition, Encounters with the 1930s. As former students of modernism and the avant-garde, this was clearly going to be the main attraction for me and the man in Madrid, and what we saw was fantastically fulfilling. Not only are the pieces on display wonderful examples of the era's experiments with abstraction (including some Kandinskys, Picassos and Calders; Man Ray's advances in photography; the jazz sounds of Len Lye's early animation, A Color Box; and a personal favourite, Moholy-Nagy's Construccion) but the exhibition's rhetoric and the depth of information provided was rich and intellectually satisfying. Aiming for an "episodic" rather than chronological view of the decade, it follows the development of Cubism, Surrealism and Dadaism through sections such as "Photography, film and posters"; and "Exhibition Culture" (probably one of my favourite sections for its unusual subject-matter, including floor plans, posters, and architecture from exhibitions and world fairs, examining these as privileged spaces of creativity and ideology).

The Reina Sofia in its entirety took more than two trips to get through – and it is worth exploring thoroughly, as I had to hunt about a while in this beautiful 17th century hospital building to find its best Dalis and Miros. The good news is that the rest of Madrid's biggest galleries are all within a ten-minute walk – but I definitely needed a tapas break or two to keep up my energy.

Also worth a look while you're here is the Caixa Forum – not a place I'd ordinarily have visited, but my architecture-loving Irishman forced his hand on the short walk from Reina Sofia. The building is stunningly modernist: designed by Herzog & de Meuron (most famous as architects of the Tate Modern) in oxidised iron, and the building facing it is an enormous art installation of hanging plants. Its current free exhibition, Towers and Skyscrapers: From Babel to Dubai is a more lighthearted display, including some impressive models of skyscrapers and original photographs, ranging from the Eiffel Tower to the Burj Khalifa, the tallest building in the world. 

Our next stop was the Museo Thyssen-Bornemisza. If you're pushed for time, the Thyssen is probably the one art museum I would leave out: though its collection holds some big names, including Picasso, Lichtenstein, Hockney, Degas and Cézanne, it has a little too much of everything and not enough of anything. The result was a scattered survey of art from ancient to modern, and its examples are not always the choicest. Their current Gauguin exhibition has some pretty paintings, but a lack of critical discourse was its weakness. In a show full of Tahitian nudes, it's all too obvious that the curators neglected to address the sexual or gendered aspects of his paintings.

The Museo Nacional del Prado is the unmissable crown of Madrid's big galleries: like most art museums of its size, you will likely spend a good few hours savouring its treasures. It's free in the evenings – perfect for bridging that Spanish gap between late lunches and their even later dinners. The Prado is as big as (or perhaps even bigger than) London's own National Gallery in its collection of masterpieces, from Hieronymous Bosch's The Garden of Earthly Delights, to more than 140 Goya paintings, including his frankly terrifying "Black Paintings", made after he went deaf: Saturn devouring one of his sons is surely the most horrific of these.

Even with a few days in Madrid, we barely managed to get through all of its biggest art institutions. There are many smaller museums and galleries I have marked to come back to next time – but I left certain of my favourite gallery, the Reina Sofia, due to its fantastic modern art collection and its commitment to in-depth discussion. All of the galleries are extremely visitor-friendly and include both English and Spanish text, and most have at least a day or two with free entry to the temporary exhibitions – definitely something I'd like to see more of back here in London.