Visiting Blenheim Palace Via Train

From the Palace’s website:

Booking a taxi in advance can provide the cheapest, most convenient journey. Visit townhousewoodstock.co.uk, minicabit.com or Woodstock Taxis to plan your trip.

From Visit Woodstock website

Getting to Woodstock from Oxford
Taxis are available at the railway station in Oxford, but there is also a frequent and convenient bus service from the station forecourt directly to Woodstock.
Stagecoach S3 service will show Woodstock, Charlbury or Chipping Norton on the front. You can find a timetable at www.stagecoachbus.com/timetables.
Oxford Bus Company operates its 500 service from Oxford Main station, via Oxford Parkway to/from Woodstock.
The bus stops are close by the Hensington Gate entrance to Blenheim Palace and in the centre of Woodstock itself close by the Marlborough Arms Hotel.
Just let the drivers know where you want to get off and they will be happy to keep you informed.

Definetly Oxford.
Frequent train from London Paddington.
I've done that trip and that's the route. Hanborough is a little country station with no facilities. I don't even think that walk is safe- seems to be down a main road which is unlikely to even have a sidewalk.
The same bus as you used to Blenheim goes through Bladon for Churchill's grave.
This is the route S3 timetable, which is the bus which runs from Oxford Station-
https://tiscon-maps-stagecoachbus.s3.amazonaws.com/Timetables/Oxford_Timetables/S3_current.pdf
This is the route 7 timetable which runs from the centre of Oxford (you would use that if you had gone to Oxford on the Oxford Tube- the high frequency luxury express bus service from London Victoria).
https://tiscon-maps-stagecoachbus.s3.amazonaws.com/Timetables/Oxford_Timetables/7_current.pdf.
This is the Tube timetable, but the train is the faster way to go.
https://www.oxfordtube.com/uploads/oxford_tube_from_20_june_22.pdf

Thanks everyone for the tips. I've worked out a plan of action, and I'm ready to roll. Monica