From London Bridge, hop on a Dartford Train via Lewisham and after a wee while you'll end up in Blackheath. Funnily enough, The Railway isn't too far from the train station. Exit, turn right, and voila.
Lewisham Station is 2 miles - take route 54, 89, or 108. Or take bus route 89 from central London, which stops in Lewisham and New Cross on the way down. There's regular service and the bus stops are right outside our door.