Oxford?

Take the train to Oxford from London. I believe that Oxford has limited access and parking for cars. Blenheim is only 10 miles from Oxford take a Bus. Oxford itself has plenty to see itself with the Ashmolean Museum, which is World Class, etc. There are plenty of Guest Houses in the area. If you want to go to Highclere; you need a reserved time ticket. Maybe get a tour package for it. While in Oxford, I went to a Shakespeare Play in the cloister area of one of the Colleges; I found out about this by having a dinner at a local Hotel. Renting a car just for 2 days may seem expensive after paying insurance, etc. From London: Hampton Palace makes a great day trip, you can boat there on the Thames and Train back. Also Greenwich is interesting with the Maritine Institute and the Guide Meridian. Bon Voyage!

If you will already be in London, I agree - just take the train to Oxford and then bus to Blenheim.

I haven’t done it (but have looked at it) and Highclere doesn’t look easy by public transit, so a car might be useful.

Unless you are truly set on Blenheim or Highclere, I would spend your 2 days just staying in Oxford and seeing all its wonders. It truly is a gorgeous city with incredible architecture, history and beautiful gardens. And don't miss the Ashmolean Museum, which is one of the most interesting museums I've ever been to.

And I agree with the others about sticking with public transport. You do not need or would want a car in Oxford.

If you're looking for an inexpensive hotel in Oxford, the Premier Inn Oxford City Center might fit your budget. Premier Inns are good hotels for those on a budget, yet are consistently clean with very comfortable beds. I've stayed at a number of them and have always been happy with the room and service. PIs provide breakfast at an additional (reasonable) cost. One thing to keep in mind is that the earlier you book, the cheaper the price. https://www.premierinn.com/gb/en/hotels/england/oxfordshire/oxford/oxford-city-centre-westgate.html

Oxford council has more-or-less declared war on the car. The one way system is convoluted and parking hard to find and very expensive

Parking in Oxford is terrible! Take the train to Oxford, if you drive to Oxford, you will have to park outside the city.

Driving and parking in Oxford is awful like most smaller towns and cities (Winchester, Cambridge). The train service is excellent in the UK and is a better way to travel. Driving from London to Oxford would take more time than the train and be more stressful. Even taking a taxi instead of the bus would be cheaper than renting a car. Many sites give directors for train and bus travel and indicate if there is a walk at the end. They may even recommend when a taxi is a good idea.

The public transportation services in the US are not very good in many places outside big cities with subways. Americans are use to using their cars are as the main transportation method. The UK, and most of Europe, have great transportation links. They are run efficiently and on time. They are often a better way to get around unless you are spending time out in the country away from towns and cities, and even then there are buses that go from village to village fairly frequently.

I don't think anybody has mentioned the Oxford emission charge which is due on all vehicles in the centre, payable electronically

Traffic in central Oxford has been a documented problem since the middle of the eighteenth century and is yet to be solved. The solution taken back in 1771 included widening some streets by pulling down a number of buildings, including parts of some of the colleges and churchyards, replacement of Magdalen Bridge, and making killing of cattle in the street illegal. These had but limited effect and the tail has been chased ever since.

making killing of cattle in the street illegal

sheep was ok?

Oxford Mileways Act 1771:
And be it further enacted, That if any Person or Persons shall kill, slaughter, singe, scale, dress, or cut up, any beast, Swine, Calf, Sheep, or other Cattle, or cause the same to be done, in any open or public Street, Lane, or Way, within the said University and City of Oxford, or the Suburbs thereof, or in the said Parish of Saint Clement, or hang up, or cause to be hung up, any Beast, Swine, Calf, Sheep, or other Cattle, or any Part or Parts thereof, in any of the said Streets or publick Passages, from and after the Market herein after-mentioned shall be completed, every Person or Persons so offending shall, for every such Offence, forfeit and pay the Sum of Ten shillings

no sheep then

10/ quite a fine 250 years ago

Haha! You guys are the BEST!!!
But seriously…thank you for all that great input.
We will take the train.
I think we will pass on Highclere, and focus on Oxford and Blenheim.
Next time, Downton!