There are a few options. This is what I did which was easy AND a cost savings.
Take the train (Great Western Railway) from London Paddington to Oxford. Take the S3 Stagecoach bus which you can take directly from the Oxford train station. Just take the walkway over the tracks for the bus.
In fact, ask the S3 bus driver for the 30% off ticket to Blenheim Palace which you cannot enjoy if you buy online in advance. The bus driver accepts cash and contactless Visa. I'm not sure about other contactless cards as I only use the Visa card.
https://www.stagecoachbus.com/promos-and-offers/oxfordshire/bus-to-blenheim-palace