There are 4 major airports in London that you should familiarise yourself with - Heathrow, Stanstead, Luton and Gatwick. I believe that there is also London City airport but I have never caught a flight from London City. You can be sure you will be spending a lot of times trying to figure out how to get to all the different airports.
Heathrow airport is the only one that has access to the tube via the Picadilly line. There are special airport trains - Heathrow Express, Gatwick Express and Stansted Express. Be aware that the price of fare can be expensive so book in advance.
There is also regular non-TFL train that you can use to get to Gatwick and Luton. Look on Trainline
to find and book tickets online
National Express does coach transfers to ALL airports and this is the most cost efficient way although it could take the longest.
Uber or Taxi Hire. This is the option I usually go with if I am travelling in a group or if I have lotsa luggages or if I'm arriving/departing at weird hours which happens a lot on the budget airlines. Tickets for the <Insert Airport Name> Express are not cheap when you are buying for two and saves you the hassle of having to drag/carry your luggage which is traumatising because not all stations are access friendly.
I also recently started paying a little extra £££ to get priority fast lane through security and would HIGHLY recommend getting it for Immigration. I have spent hours waiting in line to cross the border as I don't have an EU passport and most airports now offer this service but remember to BOOK IN ADVANCE
as you cannot purchased it on the day.Tip!!
If you are looking to book for an Airport Transfer, I would recommend TwelveTransfer. I often book with them instead of using Uber when I catch an outgoing flight to give me peace of mind that I know the car will come at 4.45am in the morning.