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.
Lovely space in Southbank. Interesting events and exhibitions. I really like the surroundings as well, especially during sunny days. Walk around and then have a rest drinking facing the river.