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.
The oldest public art gallery in London also is my local gallery in my neighbourhood. Opposite to the Dulwich Park where I walked with my puppy regularly. They have great collections of Dutch paintings as well. My husband found the original Dutch painting of the print at his home he was growing up with.
Epping Forest is a 2,400 hectares area of ancient woodland between Epping in the north and Wanstead in the south, straddling the border between Greater London and Essex. I tend to get off at Loughton Tube Station (Central Line), which is about 10–12min walk to the forest.