You can take a bike on nearly all trains and the Overground (except at peak times). You can also take it on near-surface tube lines (except at peak times). You can Google the tube 'cycle' map of these lines (Circle, District, Metropolitan ... ). You can't take a road bike on the DLR, trams and buses.
So yes getting a bike round London by rail is mostly easy outside peak times. For peak times and deep tubes - get a Brommie.