You can get a travelcard for central London (zone 1-2) for under $6, for travel after 0930. It's good on the tube and local buses. London is a great town for walking around, or popping in and out of the Tube at various locations.

Suggestions for a first timer:

At least one museum (V&A, British Museum, Imperial War Museum)--figure on return trips to eventually get to them all

Cabinet War Rooms

Visitors Galley at Parliament (I've always gone in the afternoon around 3-4pm)

Theatre (half price theatre booth in Leicester Square)

Changing of the Guard at Buckingham Palace (does not happen every day)