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.
The Design Museum is a museum in Kensington, next to Holland Park, which covers product, industrial, graphic, fashion and architectural design. There is also a nice cafe on the 1st floor with beautiful views of the park.
I've only just discovered this fantastic little salad bar-come-florist right round corner from my studio. It serves freshly made salads that are not only delicious but cheap too! Either take away, or grab the one and only table and eat in.