The thing I love most about London is the juxtaposition of urban and rural, and nowhere is this better illustrated than in Lea Valley Park. A genuine wilderness where I’ve seen everything from ponies, to a rogue Costco outlet, and all less than 20 minutes from my front door.
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.
Slightly biased as this is where I live, but Stoke Newington is great place to spend lazy saturday / sunday afternoons getting sloshed with friends. There are 10 + pubs within walking distance who all are great places for one reason or another. I'll pick out the White Hart for its huge beer garden...