When I lived in Mexico City, I had three hobbies: riding the Metro, taking turkish baths, and visiting the National Museum of Art.
Mexico, being the biggest city in the world, has a number of famous museums, including the Anthropological Museum in Chapultepec Park, and the workshops of Diego Rivera and Frida Kahlo, which all gringos must visit. The National Museum tends to be less renowned (it's hidden behind two architectural masterpieces, the Bellas Artes building and the Palacio de Correos). Beneath the mantle of fine art, the National Museum is secretly a radical place - dedicated to showing how visual art developed in this great country, and how - in the aftermath of the Spanish invasion - Mexican artists held their ground and developed their own, original voices, first in the context of religious art, then in the face of "commercial" tastes dominated by Spain, France, and European money.
The "Christian" art veers quickly towards the Virgin of Guadalupe. (The basilica of the Virgin in Mexico City is the only Christian church I've seen with a big portrait of God in it: God as in the angry-looking, berobed and bearded old man variant. The Church is terrified of the goddess cult which, not-so-secretly, fuels Mexican reverence for the Virgencita. And one big picture of a pissed-off patriarch is quickly overwhelmed by hundreds of thousands of images of the Virgin, all made in China, on sale in the vast religious tanguis which surrounds the basilica).
In the 18th and 19th centuries, paintings with "historical" themes, depict Spanish atrocities against the native inhabitants: A priest prays at the scene of a massacre, a woman clinging to him; conquistadores torture native kings.
I started visiting the museum during the big homenaje to Jose Maria Velasco in 1993. This was the biggest assembly of Velasco's paintings ever, and if you aren't familiar with his work the National Museum is still the place to find the best selection of his works. Velasco was a landscape painter, a student of Italian landscape painters who surpassed the work of his masters. I used to sit for hours in front of his Valle de Mexico desde el Cerro de Santa Isabel, a panoramic view of the valley with its lakes and volcanoes, painted in 1875, immersing myself in that luminous, unpolluted landscape, imagining how a modern painter would depict the same scene - hills and valley choked with houses, highways, skyscrapers, the lakes vanished or diminished, the air thick with diesel and benzine...
Velasco's vision may seem romantic now, but he painted what he saw, and at the time the very act of doing so was revolutionary. Previously Europeans had come to Mexico, to paint it (as they would later come to film it!), while the Mexicans held their bags, and cleaned their brushes. Velasco was a student of Landesio, and one of Landesio's Mexican landscapes hangs in the same gallery at the Museo Nacional. It is worth studying, in the light of Velasco's later work.
There is so much good stuff from the late 19th century - including an outrageously perverse statue, Malgre Tout, in which an innocent naked woman is chained to a rock as the tide rolls in - that one is inclined to linger here. Particularly in the vicinity of Malgre Tout...
But there the beginning of the 20th century, the Art of the Revolution: the touching paiting of a PRIista schoolteacher, waiting to be assigned his school; nightmarish images of politics in El Demagogo, of loss in La Revolucion. Today the Museum also has a very tony store where you can buy posters and books, and usually there's an audio-visual booth sponsored by an evil TV company or phone billionaire, playing extracts from classic Mexican films - often directed by that magnificent honorary Mexican, Buñuel.
The National Museum is quite a fine building in itself, and should you find yourself in Mexico City with a few hours to spare I recommend a trip to the Museo Nacional followed by a baño turco at the Baños Colonial around the corner. Both are accessible via Metro, even from the Airport, if you're adventurous.