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.
I've had a love affair with the BT Tower for some time now. Unfortunately entrance to the public was shut in 1980, but earlier this year I was invited up to the top, where the revolving restaurant once stood. The views were absolutely breath taking.