If you're staying in South London, always try to fly in through Gatwick instead of Heathrow. It's way closer and often much more inexpensive. There's also a train that goes right from the airport to most neighbourhoods so you can save money on an Uber.
Also no need to buy train tickets - you can use your credit or debit card (as long as they have tap feature) to pay. The transit system has a max per-day cost, so you can continue to use all public transit without overpaying as long as you tap the same card.