Regarding museums in both cities:
In MAD, three of the world's finest museums are all within walking distance to each other - Thyssen Bornemesa (sp?), El Prado and the Museo Nacional Centro del Arte Reina Sofia. In BCN, go and see the Picasso Museum, The Miro Museum, the Contemporary Arts Museum and Gaudi's architecture, especially La Sagrada Familia, which is a cathedral in continuous construction.

For shopping in both cities:
In MAD, you might want to check out the Salamanca district. In BCN, you might check out Passeig de Gracia and Avenguda Diagonal. Both are shopping streets in the city.

Getting around:
Both cities are highly walkable in spots of each city. BCN has the incredible La Rambla and tree lined avenues that will rival Paris anyday. Madrid is more classical, in fact Baroque in its urban design. Public transportation is good in both cities. MAD is more like New York. You either take the metro (subway) there or the bus. In BCN, the neighborhoods and centers are more spreadout like Los Angeles. You can get around with bus, metro and tramvia. BCN also has an interesting gondola system as well as a telerific (sp?) or funicular system to get you up high into the mountains.

Regarding something about language that I found to be progressive is that the City of Barcelona has passed a decree or a city ordinance a couple years ago stating that Barcelona has two official languages: Catalan and Castellano. The ordinance also states that all other languages spoken must be tolerated in the city. Very interesting.