In my opinion, cycling is by far the best way to travel and cummute in London. Not only does it save you a fortune in travel expenses but you get to your destination on your own terms, not crossing your fingers and hoping a train doesn't break down or your bus turns up when it's supposed to. Sure it has it's obvious dangers but if you treat other road users and pedestrians with respect, cycle legaly and smartly and wear the correct gear - Helmet, gloves, etc. you should be fine. With the recent surge in Cyclists in London and the great designated bicycle routes, cycling through London has never been better, just remember to pack some waterproof clothing.
I use my bike predominately for commuting purposes but occasionally I might take an unknown route, purposely to get lost and have a mini adventure on my way home. If getting lost isn't your thing, I found this great website listing loads of great cycle routes Cycle-Route
Enjoy, wear a helmet and be safe