It really depends on what your learning style is, Josh. Any ideas?
I'd make a start by riding along the nine most important major A-roads first. A1-4, A10, A11, A13 and A40 all start reasonably close to St Paul's Cathedral. The A2, A11 and A13 are furthest away; the A2 starts at Borough and the A11 and A13 at Aldgate. The A10 and A3 start at London Bridge. The ninth is the A5, which starts at Marble Arch. The A40 should run all along Oxford Street, but is in fact interrupted and doesn't include much of its length. London's layout originated with these nine roads, and once you know them well, you can find your way virtually everywhere else with some ease.
There is a second layer of A-roads that are worth knowing: A501, A41, A104, A3211, A205, A23, A315, A402, and A404. The A406 North Circular Road is also worth knowing. Not that pleasant, but often the quickest route around North-west London.
For those who are laughing now about knowing A-road designations--those are the ones worth knowing as they apply over reasonably long distances and save you from having to learn too many of the regularly changing street and road names. Most of the rest of London is a morass of confused, confusing, and, frankly, superfluous A-road designations. Signage for these major A-roads is surprisingly good once you start paying attention to it and can be very helpful in navigating. Like bus numbers, there are also usually some relationships between A-road numbers that can help with guessing if you're lost.
There are some roads and streets of such obvious character, like Park Lane, that I haven't included, as they're easily knowable. The above A-roads, however, include quite a lot of London's most important and distinctive streets, anyway. All the rest of London you will be able to fill in over time.
But, as I said, if you prefer not to learn from a more or less systematic beginning, another approach may work better for you. I never took the systematic approach, either, I've just been riding my bikes around London for fifteen years. I'm sure that learning about these A-roads would have made learning much quicker for me, at least. If you want to have fun, just get lost deliberately and find your way back without an A to Z, then spend hours at home desperately trying to work out where you've been. :)
You wouldn't perhaps be German would you?
I was going to say something very similar, but couldn't be bothered ;)
You wouldn't perhaps be German would you?
I was going to say something very similar, but couldn't be bothered ;)