Blenheim Palace

What do you mean by best way? The best way to get there? Or the best way to see the palace?

I have been there twice. I see they have added several Churchill exhibits which were not there when we went.
A member of the family who lives there, royalty, spoke to us all about the dukes, duchesses, and I asked about Churchill. My question was brushed aside as Churchill was not a royal! At least not 100% with his American mother.
Honestly, I don’t like touring old palaces with one exception, Buckingham Palace.

Check to see if they have any special tours on the day you visit. I think some can be reserved in advance, but I booked after arrival. I took a garden tour and a ”downstairs” tour and really enjoyed both.

Quite near Blenheim , is the church in Bladon , Very worthwhile for a reflective pause , - the Churchills are buried there

Can you expand the question? Best way for what?

If you mean the best way to get there- it is either drive - about 30 miles, or a taxi.

By transit it is Stagecoach Bus #1 or #2 for a 30 minute journey to Moreton-in-Marsh station then hourly train to Hanborough Station then walk either through the grounds or via Bladon village.
Alternatively stay on the train to Oxford, then very frequent public bus to the Palace.
A little bit messy, but not that difficult.

We stayed at the Volunteer Inn at Chipping Campden and drove to Blenheim Palace.

The Blenheim Website has a a lot of useful information regarding tours and pricing.
Though it is expensive, we enjoyed our day there very much last year - so much so that we returned another day since the tickets are essentially annual memberships. Allowed us to visit areas that we missed the first time around, such as the extensive gardens and the pleasant walk over to St. Martin's Church in Bladon to visit Sir Winston's grave.
Plan to make a full day of it - it's a sprawling, beautiful, and interesting place with a fascinating back story - especially if you're into Churchill lore.

The Marlboroughs are not royalty and I doubt they ever suggested they were.

This article should help allay confusion about this - https://en.wikipedia.org/wiki/British_nobility Also , if you visit Bladon , one of the burials would be of significance to American visitors - Consuelo Vanderbilt Balsan , who had been married to the ninth Duke of Marlborough in a loveless , arranged marriage in 1895 has an interesting tale herself . One of the " Dollar Princesses " who married into the British Nobility . her life goes from New York and Newport , Rhode Island . to England in a manner reminiscent of " Downton Abbey " https://en.wikipedia.org/wiki/Consuelo_Vanderbilt

Thanks all!
There is so much to see and I don’t want to miss anything. It seems that in addition to main tour I should add Dukes Private Quarters and Servants Quarters, walk gardens, see Winston Churchill Exhibition, and walk to Bladen. A full day!

We had wanted to avoid driving, but we are having second thoughts. Trains are expensive and sometimes stations are not near our lodging. We had planned to take train from London to Cotswolds, then to York, then to Edinburgh.

We are investigating BritRail Pass. Have any of you used it?

Driving allows you a degree of freedom and flexibility that is very useful in rural areas , I've made a number of extended trips to the UK , and outside of major cites always used a car . I'll be back this coming Autumn and have a car booked for a month of my trip for northern England and Scotland . Without a car ,many of my choices of places to see would be extremely difficult , if not impossible . to access

For the OP- trains are not expensive if you book ahead. Brit Rail is not worth it-
London to Moreton in Marsh (Cotswolds) is no more than £17.60
Cotswolds to York is no more than £57 if you book it as a Moreton to Birmingham ticket, and a Birmingham to York ticket.
York to Edinburgh is no more than £44 booked ahead (you can easily bring that as low as £25).

If your lodgings are away from stations then BritRail isn't going to solve that

I'm with Steven. Renting a car will provide the maximum of flexibility with the minimum of hassle.
It'll be worth it.