Battersea Power Station will have its own Zone 1 Underground station. Until then, the best options are:
For Circus West Village (via Grosvenor Arch)
Sloane Square (Zone 1, Circle and District lines). A 12 minute walk via Ranelagh Gardens and Chelsea Bridge or a five minute bus ride on routes 452 and 137.
Victoria (Zone 1, Circle, District and Victoria lines). A 20 minute walk via Buckingham Palace Road, Ebury Bridge Road and Chelsea Bridge or a ten minute bus ride on route 44. Victoria is also a major railway terminus and bus station with a dedicated express service connecting to Gatwick Airport.
For the south and east of the Battersea Power Station site
Vauxhall (Zones 1 and 2, Victoria line). A 20 minute walk via Nine Elms Lane or a five minute bus ride on routes 156, 344 and 436.
The nearest bus stop to the entrance at Pump House Lane is 'Sleaford Street'. The following buses serve this stop: 156, 344, 436. From here, cross the road to the entrance at Pump House Lane, and walk along the pedestrian route to Circus West Road (10 mins).