London is a big city and it can take some time to get from A to B. Here is some advice on the fastest and cheapest possibilities of getting around.
1. Ride a bike!
There is no cheaper (and no faster) way to get around London than using a bike. The city bikes are available all around the city. They are not expensive and every ride for half an hour is for free. If you are staying for some time in London you even might think about ordering your personal “key” otherwise you can just use your credit card at the stations.
Other than that you might consider to buy a proper bike yourself with a decent lock (!). This has the advantage that you can sell it again before you leave. You can buy used bikes on Bricklane Market every Sunday, or check out Gumtree. However, my reccomendation is a small open community bike shop called Cycle Pitstop. They have great bikes and they´ll give you a good price.
By riding through London, you will get to know the city better than any local and experience London life better than ever before. There are even a few bike highways, which are worth to be checked out.
Don´t forget to always ride very carefully with open eyes.
2. Use the Oyster Card!
It might be obvious, but if you don´t know about this public transport card, you´ll pay very expensive prices for getting around by train, subway or bus. If your credit card works with the contactless paying method, you can also use it and it will deduct the same price like an Oyster Card.
If you are a tourist and want to enjoy a typical London bus ride, don´t get into the expensive tourist busses. You´ll get a more authentic experience with the normal public transport. Hop in, pay with your card and go to the upper level. If you are lucky the front row seats are not occupied. Online is enough information about good routes. Here you can find three recommended rides by the Visit Britain Shop.
3. Use Uber!
We have all been in the situation of being stuck somewhere in the middle of the night. Or you are moving and just need a way to transport some heavy suitcases. Of course you can just get a normal London Cab. However, cheaper, better, faster and more transparent is Uber. You can order your ride via Smartphone, see the price in advance and additionally check, where the car is right now. To get your first ride for free follow this link in order to sign up to Uber.
Uber works in different countries all around the world, so you might be happy about your app even in Colombia, the US, Peru and so on.