USING THE LONDON UNDERGROUND, AKA "THE TUBE"
We strongly suggest that you purchase a Tube and Bus Pass - which is called an "Oyster Card" - for travel during your time in London. You should buy a Pay As You Go card. You'll put a "deposit" on your Oyster Card and receive a plastic card which is read by card readers as you enter and again as you leave each station. The card reader will tell you how much money you have left on your card. When the amount is depleted, you can use a machine and a credit card to "top up" the Oyster Card. The card is valid both on the Tube (the underground) or on the innumerable buses in London.
Paying cash for each ride will cost more than double the price (about $8 per ride as opposed to $3.00) and you'll also have to stand in lines to buy tickets. A single tube or bus ride using your Oyster Card costs £1.50 ($3.00). You can purchase your Oyster Card either at the airport or at a Tube Station.
It's possible to purchase your Oyster Card for use of the London Underground from the U.S. and have it delivered before you travel. The website is http://www.visitbritaindirect.com. Just click on the right column to indicate you're coming from the U.S. and you'll see the information about the Blue Oyster Card. You can put whatever amount of money you wish on the card and when you've used that up, you easily can add more.
There's a shipping and handling fee of $10.00 which you'll avoid if you wait to purchase the card in the U.K. You can decide if the convenience of being ready to jump on the Tube at Heathrow is worth the price.
Oyster Card information plus a printable Tube map is available from http://www.tfl.gov.uk/tube.
For additional information about the Tube, check out http://www.visitlondon.com/tubeguru.