Depending on where you are going, the Heathrow Express really is the fastest and least expensive route to central London. While you may be dying to ride in a London taxi (which no doubt is very comfortable), take the Heathrow Express.
Follow the signs to the Heathrow Express - you can't miss them.
You need to buy a ticket - the machines are very easy and take most major US credit cards.
Buy a ticket.
Head down the escalator or take the elevator down to the departure gate. It's a 5 minute walk once you get off the elevalator. The longest you will wait is 15 minutes, and the ride takes about 15 minutes. The trains can be delayed but this rarely happens. The train will let you off at Paddington Station.
