You can take either the 209, 283 or 419 from Hammersmith Bus Station which you will find on the ground floor level next to the Hammersmith Piccadilly and District Line Tube Station/Shopping complex.
The 33 and 72 routes also serve parts of Barnes, crossing Hammersmith Bridge before passing down Castelnau and across Barnes Common. The Red Lion stop on these routes serves the Wetland Centre.
If there's one available, the best bus Barnes Wetland Centre is the 283 which occasionally drives direct to the centre and no further or alternatively stops just before the Red Lion pub close by.
The 209, 283 and 419 routes turn right at the bottom of Castelnau to go through the village and towards the pond and the river.
Take the 419 from Bus Stop B on George St
Find more information and useful travel updates go to the Transport for London Journey Planner
TfL also have a live bus arrival map here so that you can see how long it will be until the next three buses arrive!