From LHR, it's the Piccadilly Line (the blue one). Click HERE for the map. LHR is at the bottom left corner.

You can either get on at the terminals or at Hatton Cross, which is right next to the BA engineering base. For the central London sights, get off anywhere between Green Park and Covent Garden. If you can get hold of a tourist map, you'll see most stuff is relatively close by although the nearest station for the museums is South Kensington.