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.
Being down on the underground can be quite stressful and draining, but if you have the chance to look up and into the details that has gone into the tiling of the platforms and stations you might start to see some beauty down there. A lot of stations has bespoke tiles and decorations, almost a century old.
Bethnal Green station is one of a handful in London to have been given a very specific additional decoration to the classic cream tiles and name strip. Easy to miss, but dotted around the station are a series of tiles with raised motifs on them, representing aspects of London and places that the Underground visited.
Or for typographers: check the type on Hampstead station or Holloway Road for some inspiration.
London is so vast I end up spending hours travelling on trains. They are by far the best way to travel around on and are always full of weird and wonderful characters.