At first the public transportation system in London can look very complicated and daunting, however once you are actually on a bus, tube, or a train, it is actually quite simple. The first thing that you should do is get an Oyster card which can be done at many shops in London as well as train or tube stations. This may not be possible as soon as you land, however, it is a cheaper and easier way of traveling around the city.
Once you land at one of the London airports, it is relatively simple to get around from each airport to your accommodations for the trip. I would suggest knowing which buses, tubes, or trains that you should take and which stops you should get off at before landing to make the trip from the airport less stressful. An excellent site to help plan your trip is http://www.tfl.gov.uk/, in the top right hand side of the page there is the Journey Planner. With this you can choose the start and end location of your trip as well as the time and day that you are leaving on the trip, for example, the time and day your flight is scheduled to land. Travel for London also has more detailed information about how to obtain an Oyster card as well as topping them up and how to use them.
If you have any trouble while trying to get around on the tube or trains just simply ask one of the workers at the station and they will be able to help you figure out which line your stop is on and which way it is going. The most important thing to do is keep calm.