Public Transport – There are two ways to go here: the Oyster Card or daily/weekly travelcard. Oyster works using top-up so you add money to it as you go along; travelcard you can buy daily or weekly. Im happy to discuss this with you before you visit.
From the Airport - From Heathrow you can either take the Heathrow Express into Paddington which takes about 15 minutes or the underground into central London, which takes about 45 minutes. A taxi will normally cost you in the region of £70-80 into central London depending on traffic and take anything from 30-45 minutes, again depending on traffic.
From Gatwick there is the Gatwick Express to Victoria station which takes 30 minutes. A taxi will cost in the region of £120.
Taxis – The black taxi drivers (some of the taxis are now blue, green, red and a variety of colours) are the best in the world in my view. They do a test called ‘The Knowledge’ and it can take from 3-5 years to pass, so they know their stuff. These can be hailed on the street.