Trains, buses and taxis
London Paddington - Reading - Oxford local services, operated by Great Western Railways, call at Pangbourne Station, which is 5 - 10 minutes walk from Whitchurch-on-Thames.
Trains run about half-hourly - journey times are about 10 minutes to Reading, 30 minutes to Oxford and 1 hour to Paddington (or change at Reading and 30 mins from Reading)
See the live departure board at Pangbourne Station
Bus Service 142 - Going Forward Buses - Wednesday and Saturday only
9:59 and 13:59 from Whitchurch returning from Reading at 12:15 and 15:45.
See here for more details. This is called the Parks Route - this service runs from Goring to Pangbourne, to Whitchurch, Crays Pond, Caversham, Tesco and the centre of town and back again.
Other Going Forward bus details here.
Bus Service 143 - Thames Travel
Bus service between Pangbourne and Reading along Oxford Road: the timetable is here.
Here is a useful link which gives a list of registered taxis, insert your postcode and it will give local taxi firms.