Since you will probably be an exhausted zombie after your flight, it's best to plot your exact route to your hotel (etc.) ahead of time. Write it down ahead of time. The less you have to think at the airport, the better for you.
On the plane, you will be given a landing card for passport control that will ask what address you will be staying at. So make sure you have that written down, too. If you are traveling around the whole of the UK, it's ok just to put the address of the first hotel.Heathrow Airport:
A taxi from Heathrow to central London will cost you £60-£100 depending on where you are going and the traffic along the way.
The Heathrow Express train is overpriced and won't really save you much time unless your hotel is walking distance from Paddington Station. And even then, it's probably ok to take a non-express train to Paddington which will be cheaper.
The Underground from Heathrow will take about an hour to get where you are going (less if you are staying at Earl's Court, more if you're staying in an outlying neighbourhood) and is 'inexpensive' compared to other transit from the airport.
Although it is cheaper than other methods of transit, the Underground still isn't cheap compared to the Metro in other cities. Deal with it. Welcome to London.Buy an Oyster card!
Let me repeat that: BUY AN OYSTER CARD!
Unlike most travel cards in other cities (looking at you, Seattle!) Oyster cards give you a discount every time you ride the underground or a bus or a train--it isn't just a debit card for travel. Get a card at the airport, use it! Each person in your group/family needs an individual card, you can't just buy one and share it (like you can do in Barcelona).
Oyster Travel Pass vs. Pay as you Go. If you are staying in London for a full week (7 days or longer) and plan to travel by Underground or Train every one of those days, then buying a travel pass may save you money. If that is not the case (day trips out of town, rest days, doing the hop-on-hop-off tour bus one day, staying less than 7 days) then Pay as you go is probably going to be cheaper. It will automatically cap itself at the 1-day travel card rate each day, which is approximately 4 rides.
If you are visiting friends who live in London, do not ask them to meet you at the airport. If they WANT to meet you at the airport, they'll make the offer. Even if you've never used a subway/underground before, don't be intimidated--it really is quite easy. By the time your friend gets out to the airport (one hour or more) waits for you to get through customs (another hour or more) then rides back to their home with you so you can dump your luggage before sightseeing (yet another hour or more) you've pretty much wasted half their day. At most, ask them to meet you at the station near their house. And, if at all possible, call them from the airport (once you're past customs) and say you're on the way. Everyone who lives in London knows how long it takes to get from the airport to their station. And can time it so you each will arrive there within 10-15 minutes of each other.You can easily walk a mile or more through the airport after landing. Especially Heathrow. Most especially Heathrow Terminal 5. Plan accordingly.
A taxi from Gatwick will cost even more than one from Heathrow.
I've never used the Gatwick Express to Victoria Station, but I'll assume that it is also overpriced. There are lots of other (cheaper) trains to Clapham Junction, Victoria, or London Bridge which will get you there pretty fast.
NOTE: Gatwick is outside the Oyster zone, so you will need to buy a regular ticket from Gatwick to wherever you are going. The same will be true on the return trip if you leave from Gatwick.
It's fine to skip the ticket machines and go to the person in the ticket window and tell them where you want to go. You're an exhausted zombie! Even if you were in a normal mental state, the ticket person would still know more about how the trains work than you do. Let them help you.
You clearly know what you're doing and don't need my advice.
Luton or Stanstead Airports:
You're on your own...you poor unfortunate fool! Now you'll find out why that flight was so inexpensive.