I really recommend the Museo Franz Mayer, which has existed since 1986 but where I had not been before. I recommend its collections, its cloister, its fountains, its patios, its café, and last but not least its working library. This is one of the important places to know about in Mexico City and it is only a stone’s throw from Bellas Artes (Plaza Santa Veracruz, facing the Alameda on the Avenida Hidalgo side).
The building dates from 1582 when the first doctor to graduate from the Royal University of Mexico created a hospital for Blacks, mulattos, mestizos, poor Spaniards, and orphaned children. Originally it had a church dedicated to the Three Kings; now its plaza has two churches. The hospital lasted for most of four centuries, although much of the current edifice dates from the late eighteenth century. I want to visit it again. If you have a strong Internet connection, you can undertake a virtual visit.
I had walked there from the Chapultepec metro station, which is a long way. To get back I took a green bus along Reforma; these green buses commemorate the bicentennial. They cost 4 pesos 50 centavos and they are very luxurious.
Axé.
coldhearted scientist وداد 
Sounds fascinating. I may have to reconsider my boycott!!!
I don’t want to go down there as an American know nothing, like everyone else in my acquaintance (aside from you) who visits there.
It’s actually a really easy place to travel and very used to American tourists interested in culture. You can go around in a modest way — you’ve traveled in Europe, you know how. And in Mexico City you can jump on the Turibus http://www.turibus.com.mx/index.php?option=com_content&task=view&id=7
to get oriented — for $7 if you’re over 65, you ride on it all day. It has 15 stops and you get off at the ones you like, visit the area, and then hop on the next Turibus to continue.
The other museum I like for the building as well as the collections is the former Archbishopric’s palace, now a museum of contemporary art. It’s under renovation right now though, so you can’t see most of the collections, although they will let you in to the patio and corredors and you can feel the vibration. http://www.apartados.hacienda.gob.mx/cultura/apartados/museo/index.html