Just a quick thing I wrote up to advise someone else. Thought I might as well share.
Orientate yourself with a long walk. Start at Buckingham Palace, walk down the mall to Westminster, then walk along the Thames river path (either side). Gaze at St Paul's from across the Millennium Bridge and the outside of the recreated Globe theatre from the outside, nearby. If you have the energy, continue along Southwark towards the London Bridge.
Walk down Portobello road market on a Saturday. Buy lunch from one of the amazing food stands. Stop in the Travel Bookstore to browse. (http://www.thetravelbookshop.com/)
Take the lift to the top floor of the Tate modern and enjoy the view. Take the shuttle boat to the tate for a cheap river cruise and an efficient way of getting between the two museums. (http://www.tate.org.uk/tatetotate/)
There are so many parks and museums in London, all free, it is hard to pick which ones you want to go to! Don't miss the British Museum or National Gallery. Sit is near-by Russell Square or Trafalgar Square (respectively) and people watch.
Get lost in the West End. Ask at box offices of theatres for cheap last minute seats, if that's your thing. Explore Covet Garden.
Grab a curry and some beigels in Brick Lane. (http://www.londontown.com/LondonInformation/Restaurant/Brick_Lane_Beigel_Bake/a7e6/)
Attend evensong at St Paul's (http://www.stpauls.co.uk/) or an organ recital at Westminster (http://www.westminstercathedral.org.uk/vinfo/vinfo_times.html) and see the inside for free!