The cheapest option is to take the Piccadilly Line on the underground, but the quickest (depending where you're coming from) is to get the Heathrow Express from Paddington, which takes about 15 minutes.
Just take the 'Tube'. Very convenient, particularly as is Piccadilly Line, can get off at Piccadilly Station, close to Regent Palace Hotel, which we stayed at several times. Believe that Hotel has now closed. Shame, as it WAS just SO handy!