Make time for Piccadilly Circus, a tour of Buckingham Palace, Tower of London, a double-decker bus tour, visit Cambridge and don't forget the nightlife.
Quick Tips:
When I first arrived, fresh out of customs, the first thing I did was get area maps, including a map of the Tube (London’s subway). That is imperative. I highly recommend the Tower of London tour. There was a lot of historical information, guides were in period costumes, and also there were paranormal/ghost tales included.
Cambridge is a great little village - you can take a double-decker bus (whole day tour) or train out there and spend the whole day walking on cobblestone streets, taking tours or just sightseeing. Make sure you take time for tea, usually in the afternoon around 3:00 or 4:00. Shops usually close, and all the cafes have outside seating, and serve tea.
Make sure you plan an evening to go to Piccadilly Circus - it would be best to go after dark, and all the lights are more dramatic that way. You'll find the clubs will start getting busy around 9:30 or so.
Greenwich and Portobello Road are great places for shopping, they have tables lined down the road where you can buy trinkets, both new and used.
Best Way To Get Around:
Walking, Tube, train, double-decker buses and taxis. (Don't take the taxis if you don't have to, they are worse than in New York.)