I don't think there's been one single day in London that I haven't taken the Underground, or the Overground, or the DLR. The public transport in London is very advanced for such a big city. It links the extremities of the city to the center without causing too much disruption on the streets.
The underground is one of the best things London has, you travel without traffic, without getting rained on. Occasionally it gets really crowded, for example never take the central line around 6pm on a week day, you will be crushed. Another recommendation do not walk up the stairs at Covent Garden unless you want to pass out half way up. No matter how much you're in a hurry taking those stairs is never worth it.
The biggest flaw of the underground is that it closes at midnight. This means when you're out at night you have to take a bus or a cab.
It took me a very long time to get used to taking the bus, especially fully understanding which routes could take me home from different locations. I still write down what Google maps suggests when I go somewhere new. I am lucky though, I have a bus that goes from Oxford Circus to my house. I've never gone to west London by bus so i don't know how well linked it is, and I tend to not go out in Camden because I don't know how to get home.
I hate the train systems that bring you to the airports. I've had to go to Stansted and Friday I'm going to Gatwick, i usually fly to London City Airport so I really hate going to airports with the tube and trains, I feel like the tube should go to all the airports. I just don't particularly like big train stations because there aren't always all the labels showing where to get which train. The streets in London are also like this, when I first got here I didn't know my way around but it didn't help that none of the roads are labelled.
Here I've attached a map of the underground with labels of special things at each location.