Walking in London
"...to walk alone in London is the greatest rest." - Virginia Woolf
The only way that it's really possibly to grasp the (less than gridlike) street plan of London is, I think, to walk it. The London A-Z is your friend here (and if you can find the room, don't use the mini one as it has no magnified central London section). Just get to somewhere central and head off in a direction. Keep taking note of a few street signs, and see what you can see. Eventually you'll almost certainly hit another tube station (actually, in the centre, if this takes longer than 20 minutes, stop walking in alleys and find a main road...)
Keep mapping how the tube stations really match up (even if it's a link every week or two) and eventually you'll begin to grasp a mental map of the 'real' London, and possibly how to compare it to the 'tube' London that newcomers to the city navigate by. (This is particularly useful near where you live; see Adjacent Stations for more on this.)
On a less practical note, I think it's useful to reiterate Woolf's quote at the top of the page.
Even for those of us who only use the city as a weekend and evening playground, after a busy day's shopping in Oxford Street or CyberDog in Camden Town it is the perfect relaxation to walk the less than half an hour between these places and Euston Station or Kings Cross; before a night at Slimelight, the brisk walk up the hill from Kings Cross is perfect to prepare you for a night's heavy clubbing after a messy train journey -- even for those of us in five-inch heels.
To people who are not residents of a big city, London can seem to propel us with a mind of its own. Once every few minutes we make a decision, which we must take in an instant, to go along one escalator or another, to cross or not to cross a road, to dive into a shop or to stay out. Travelling in London can be a set of impulses or step-changes. This is why walking can be a spiritual rest. When we walk we propell ourselves at our own speed, have time to stop and watch pigeons feeding, or to get a kebab, to walk quicker or slower, to smell cooking and to discover old buildings and skylines. Travelling like this is not a set of impulses, but a set of continuous variations in thought, speed, apptitude and attention.
When I visited London I would often come back with a headache, or mentally shattered until I decided to walk at least a couple of miles whilst I was there.
You don't even have to have a route - why not try walking with a method, or even no method at all?
In the 1950's the French Situationists developed a technique for travel which they called the derive, the "drift". They were disgusted with themselves for never leaving the usual ruts and pathways of their habit-driven lives; they realised they'd never even seen Paris. They began to carry out structureless random expeditions through the city, hiking or sauntering by day, drinking by night, opening up their own tight little world into a terra incognita of slums, suburbs, gardens, and adventures. They became revolutionary versions of Baudelaire's famous flaneur, the idle stroller, the displaced subject of urban capitalism. Their aimless wandering became insurrectionary praxis. - Hakim Bey, Overcoming Tourism
Walking with a pattern (such as "second left, first left, third right") is part of the intriguing field known as Psychogeography.
Central London Walk
A great walk I always recommend to tourists is to catch the tube to St Paul's Station, walk down past St Paul's Cathedral towards the Millennium Bridge, cross the bridge and visit the Tate Modern. It's a great view of London and a nice way to spend a day. You can vary it by instead starting at Bank Station from where you're in the heart of the [Square Mile]?, London's historic financial district.
Guided Walking Tours
A number of companies offer Guided Walking Tours around London, usually based around some theme such as "Spies and Spycatchers" or the ever-popular Jack the Ripper. You'll often see groups of tourists on the latter tour around Spitalfields.