London is quite well connected by public transportation. Riding on big red buses is a good way to explore the city. Hop on and Hop off, whenever you find something interesting to see. Fast way will be using Underground (Tube).
RECOMMENDATION: Get an Oyster card at the airport you arrive. As it is not very easy to find tube station that will help you get a new Oyster card. At Heathrow airport (LHR), just follow sign to Underground / Tube station and there is a ticket booth where you can buy it and decide if you want to also buy a day pass to get on the tube from airport to London Centre. My experience was – it took about 45 to 55 minutes from Heathrow to Piccadilly Circus (blue line), if no sudden issue. A word of warning, if you are travel light, it is a good experience but if you have a few luggages. Better order airport transfer. I have been using this airport transfer service since 2013. Reason is sometimes (my past experience once out of twice) you will be required to change at other station, if that station does not have escalator or lift, you are stuck, unless you can lift all your luggages and walk up up and down down. The design is so intelligent that will help you exercise but not very travellers friendly (nor elderly friendly), I have to say.
So, after buying the Oyster card, just go and find your airport transfer driver that you pre-booked before leaving your home town.
Get a London tube map to find out more, when you are planning your trip. I prefer taking buses, so get a London bus map too.