Bus
The high numbered train platforms are on the middle level. If you go outside from here there is a small bus station with routes including 43 (north to Islington and Finchley), the 149 to Edmonton and the 388 to Stratford.
If you walk past the bus station and slightly downhill, there are more stops on the main road (Borough High Street) and also on the lower level on Tooley Street.
Getting to central London by bus from here is not an easy job. The few that go in that direction go around the houses, as we say, and are slow. Changes may well be needed.
Crossing the bridge and looking for a bus near Monument may be better. The 133 goes to Holborn but it does pass London Bridge station anyway.