The AA is Britain's oldest school of architecture, but anyone can take advangte of the great work going on there. You can catch really interesting public lectures and free exhibitions throughout the year. There are also occasional installations and event launches, a bijoux cafe and brilliant bookshop. A great place to hang out with the builders of the future.
In my opinion this is the best bookshop in London, it stocks all my favourite magazines and has a wide selection of photography books, I spend a lot of my time there just browsing through all the books they have. David Bailey once described the place as “maybe the best photography bookshop in the world”
The Barbican is worth visiting for it's brutalist architecture alone. The concert programming is really diverse; it's home to the BBC Syphony Orchestra, but you'll also find bands like the Dirty Projectors and the National amongst the classical and contemporary composer line up. Plenty of art, dance and theatre too, however, the Conservatory is the real gem that's not to be missed.