Eccentric and surprising house and museum of architect Sir John Soane. Mostly unchanged since his death over 180 years ago, it is full of atmosphere. Also have spotlit, late night openings.  

A walled secret garden, the Physic Garden is like a time capsule, founded in 1673 to allow apprentices of the Worshipful Society of Apothecaries of London to study the healing properties of plants, it is now a living museum. Tours are offered by volunteers with immense knowledge of botany and the cultural history of the garden or visitors can wander the grounds alone and spot odd and fascinating plants such as the mystical Mandrake.

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.

The walk along the South Bank from Westminster to Tower Bridge will take you past a plethora or interesting sights and cultural hotspots. There's always something interesting going on too.

Nice place for a cake and coffee.