Frequent trains to East Dulwich and Denmark Hill overground from London Bridge, Victoria, Clapham Junction, then about 10 minutes walk or take the 40, 176 or 185 buses.
Trains also go to West Dulwich: take the P13 from here.
40 from Aldgate or Dulwich library 176 from Oxford Circus or Penge 185 from Victoria or Lewisham P13 from Streatham or Surrey Quays
Buses from Elephant & Castle
Take a train or bus to Elephant & Castle, then a 40 or 176 bus.