This page features a list of all the places we visited and how to get there by tube. The tube stations mentioned are the stations at which you are to get down. The lines mentioned are the lines that reach the mentioned station.
I didn't include bus numbers, because I think that with buses consulting a map will be best.

The underground sign

Bus station sign.

Lines color index:

Bakerloo - brown.
Central - red.
Circle (circle) - yellow.
District - green.
Piccadilly - dark blue.
Hammersmith & City - pink.
Northern - black.
Metropolitan - purple.
Jubilee - grey.
Victoria - blue.
Waterloo & City - very light green (and a bit bluish. sort of light turquoise.)
East London - orange.

• St. Paul's Cathedral

  
  St Paul's station, The central line.

• The Guildhall

  
  St Paul's station, The central line, and then some walking.

• The Barbican Centre

  
  Barbican station, The circle line, the metripolitan line, the hammersmith & city line.

• The Tower of London

  
  Tower Hill station, the distric line and the circle line.

• Piccadilly Circus

  
  Piccadilly Circus station, The piccadilly line and the bakerloo line.

• Leicester Square

  
  Leicester Square station, The piccadilly line and the northern line.

• Trafalgar Square

  
  Charing Cross station, The bakerloo line, the northern line and the jubilee line.

• The Big Ben & The Parliament

  
  Westminster station, The distric line and the circle line.

• Westminster Abbey

  

  • Westminster station, The distric line and the circle line.
  • St James's Park, same lines. The Westminster station is closer.

• St James's Park

  

  • Westminster station, The distric line and the circle line.
  • St James's Park, same lines. The Westminster station is closer.

• Buckingham Palace

  

  • Westminster station, The distric line and the circle line.
  • St James's Park, same lines. (closer)
  And then you need to walk to the Palace.

• The National Gallery

  

  • Charing Cross station, The bakerloo line, the northern line and the jubilee line.
  • Piccadilly Circus station, The piccadilly line and the bakerloo line.

• Hyde Park

  

  • Marble Arch station, The central line.
  • Hyde Park Corner station, The piccadilly line.
  • Knightsbridge station, The piccadilly line.

• The Natural History Museum

  
  The South Kensington station is the closest, but you still need to walk. lines: the piccadilly line, the circle line and the district line.
  Another possibility is to take the tube to Knightsbridge station (see Hyde Park) and then to continue by bus (number is 74).

• The Victoria & Albert Museum
  See the Natural History Museum.
• The Sherlock Holmes Museum

  
  Baker Street station, The bakerloo line, the circle line, the metropolitan line, the jubilee line and the hammersmith & city line.

• Madam Tussaud's & The Planetarium
  See ""The Sherlock Holmes Museum".
• The London Dungeon
  London Bridge station, the northern line.