A nice fast way, is Heathrow Express, a high speed train service between Paddington Station and the airport. I like it, it's convient, comfortable, nice seats, and only takes 15 minutes travel time. If you don't mind the price, under $30usd one way, $45 round trip. (Note prices vary).
The Tube, London's Underground subway is an inexpensive journey on the Piccadilly line. It takes about 50 minutes to downtown, slightly, ever so slightly, longer than a taxi ride.
I recommend the tube, as long as you don't carry too much luggage. If you have the extra cash and like your comfort, go on the Express. Taxi are the most expensive and the way to go if you want to be taken care of and not worry as to how to get to your specific destination.
On the Tube you will see a cross section of Londoners, and bit of the country side and city buildings on 1/2 the ride. The seats are okay, the cars reasonable, not the newest, but okay.
Arriving into Heathrow.
Heathrow is nice airport.
Outside one of the Waterloo Station Underground exists (there are a few).
The Underground has various levels of moderinzations. Overall, an excellent subway system.
Underground map showing Heathrow (bottom left), Paddington Station (center top) where the Express goes to, and Waterloo (lower right).