Take a train from London Heathrow airport to downtown London. 

You have two train options.  One is expensive and fast, the other is cheap and slower. 

Option 1: Train called "London Express"

London Express is Fast, takes only 16 Minutes, costs 21.5 Pounds ($33 USD) one way...

Location:  Lower level, in the center of the airport.

Option 2:  Subway Train, 'Blue Line'

This train is slower, 45 minutes, but takes you directly downtown and costs less. 6 Pounds ($9 USD) one way. 

Location:  Same Location, Lower level, center of the the airport. 

For both of these trains you can purchase the ticket right at the station before you get on the train.  The blue line subway comes about once every 5 minutes. 

There are many stops/stations you can chose to get off at, I recommend the "Green Park" station.  This stop is right in Green Park, which is right next to Buckingham Palace and the road that leads you to Big Ben.