Short version: TFL is proposing cycle paths that are fully separated from other traffic. You can vote for the proposals here (East-West route) and here (North-South route) until the end of October 2014, more information here (E-W) and here (N-S). Let’s make London the new Amsterdam!
Plans for Blackfriars Road
Although I’m not walking to Brighton myself, I created a map to be able to find my friends on their way. You can browse the map in Google Maps.
Alternatively, if you have a GPS handheld (e.g. a Garmin eTrex or similar) or phone application (I recommend Maverick – it supports Ordnance Survey Explorer Maps), you can use either this KML file or this GPX file to import the map into your device.
London Brithon Trek 2014 (map detail)