The Almeida, in Islington, is a small theatre with big influence. It's an intimate place to discover groundbreaking theatre and get up close to some of Britain's leading actors. The cafe/bar across has a constantly-changing tasty menu, relaxed atmosphere and free wifi. Open all day and great place to spend time even if you're not going to catch a production.
I'm a big fan of Richard Serra. As this is the train station I use to pass in and out of London. I think it's safe to say i've spent more time gazing and contemplating this art work than any other on earth! The steel monolith sits in a rather petite amphitheatre with surrounding step/seats. During the week, it's packed with picnicking stray commuters. I just love sitting there.
Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.