The Right Gear
Moving around London is daunting. The Square Mile is one thing, but traveling further out allows to redefine confusion quite quickly.
Make sure you have a relatively modern, SIM-free smartphone. Get a 30-day prepaid, all-the-data-you-can-eat SIM from any of the available operators. (Doesn't really matter which one.) Install the stock maps application and Citymapper. The two complement one another beautifully when trying to find out the sanest way to move around. The small cost of the prepaid SIM will be more than balanced by the lack of headache if you're staying in the City for more than just a day or two.
Invest in a pair of really good walking shoes. Break them in before London, too. The hard streets around the city are not the kinds of surfaces you want to treat your feet to in unfamiliar footwear.
Schedule as much of the inner city travels outside peak hours as possible. The tubes and trains are packed during peaks, and the buses are an outright atrocious experience. Oversized vehicles in one of the most congested cities in the world are subject to the same traffic as everything else around there. Only more so...