If you live in London, then the cheapest way to get around is with an Oyster card. It is a pay as you go card, and is quick to use, as you just swipe. No need to fiddle around with change. You can top up your Oyster at train stations and newsagents. Seniors can also get an Oyster freedom pass for free travel.
If you are a tourist, then taking a tour bus or boat is the best way to get around, as it means you get to learn about where you are travelling at the same time.
There are a number of options which is dependent on the time you have. A cab may get you to your destination but can be pricey. The train is cheaper but might take longer. However, i prefer the bus, if you have time. You see more,