London Transport produces a series of 14 free cycling maps of London. You can get these via their web site, but generally it's quicker and easier to pick them up from your local bike shop. They're pretty good, but not worth following slavishly; most of the "signed routes for cyclists" are just normal roads with the occasional blue-and-white sign, and no better or worse to ride on than other roads.
London Transport's Journey Planner web site has a cycling mode, but it sometimes makes some very strange recommendations (and assumes you'll be cycling at an average of 11.2mph). Useful as a guide, but again don't be afraid to ignore it.
OpenCycleMap presents OpenStreetMap data in a cycle-friendly format. CycleStreets is a cyclist-orientated route finder. Both of these put rather too much emphasis on cycle facilities for my personal preference.
You are allowed to take a bicycle on the Underground, but only on some lines and at certain times (broadly, shallow rather than deep lines, and not during rush hours). See this London Transport page and this map for details.
There is no consistency among train companies as to whether you may take a bicycle or how much it may cost. This list seems to be kept fairly well up to date.
Canal towpaths are great if you're fairly confident, but don't expect to make good time; I lose about 2mph off my average speed when using them. Also not at all good when they're crowded, e.g. during commuter times.
Remember that every road is a cycle route; the wider the road, the more room you have to get out of the way of faster traffic. (And, pace some cycling activists, this is only sensible and good manners: if someone wants to overtake you and there's room for him to do so safely, let him! Then you don't have to worry about him lurking around behind you.)