UK trip

It is a VERY open ended question. The best way to get around will really depend on where you decide to go. Wales is probably best done by car. You may find it best to do a combination of train / car depending on what your stops are. I would start by getting a copy of the RS Great Britain guide book. You might get better replies if you at least list what your interests are.

maybe start here?

https://www.visitbritain.com/gb/en

And check out the Britain material under "Explore Europe" on this website.

With ten days I wouldn't try to visit all three regions. You could easily spend ten days in and around London, or exploring Scotland. Remember that every day you move you're spending hours, maybe the whole day, dealing with hotels and transportation. That's time you can't spend visiting the places that interest you most. I'd suggest spending half your time in London (maybe with a day trip outside) and half somewhere else (maybe with a car to explore nearby).

Because your question is, as you say, very open-ended, my advice would be to study a guide book or two about the UK and then come back looking for help. It would be useful to know what your interests are ....... countryside, gardens, villages, museums, stately homes, foot paths? As it is, we have nothing to go on. C'mon back, y'hear?

Thanks, all, for the good advice. Agree we have to do more research.

10 days isn't really enough time for all three places. Look through where you might want to go and then choose either Scotland or England/Wales or just England.

I recommend keeping in mind that driving distances are deceiving in the UK due to traffic congestion and more 2 lane roads.. What may be doable for a day trip in the USA might take longer in the UK. A good way to estimate driving times is by using Route Planner on the following website (British equivalent of AAA)

https://www.theaa.com/

You have posted on the Wales forum - so you may like to start research on the official tourism website for Wales:> http://www.visitwales.com

If you are thinking of heading to Wales, I would be inclined to land at Heathrow & take a bus to Oxford (or bus link to Reading station & train to Oxford). You won’t need a car in Oxford - check into hotel & walk around whilst getting over time zone change & flight.
Day 2 could be a visit to nearby Blenheim Palace - for which you may need a car. Then head off a short way west to The Cotswolds for 2 or 3 nights = Burford, Bourton-on-the-Water, Broadway, Chipping Campden (with Hidcote Manor Gardens). You may even have time to nip up to Stratford-upon-Avon.

Then I would leave the Cotswolds for Wales - head for Tewkesbury and join M50 for journey into Wales.

Wales is amazing! We love Wales' natural beauty (coast/valleys/mountains), castles ("Wales has more castles per square miles than any other country in Europe!"), architecture, people, and food. Here is some information to help you plan your trip.

Itineraries
http://cadw.gov.wales/Traveltrade/Itineraries/?lang=en

Things to do / Attractions to Visit
http://www.visitwales.com/things-to-do

Underground Tour at Big Pit National Coal Museum
https://museum.wales/bigpit/tour/

The Welsh bucket list: 100 things to do in Wales before you die
http://www.walesonline.co.uk/whats-on/whats-on-news/welsh-bucket-list-100-things-9656322

Dark Sky Reserves in Brecon Beacons and Snowdonia
http://www.darksky.org/idsp/reserves/

Top Things to Do in Wales
https://www.tripadvisor.com/Attractions-g186425-Activities-Wales.html#ATTRACTION_SORT_WRAPPER

Top Reasons to Go to Wales
https://www.fodors.com/world/europe/wales/experiences/top-reasons-to-go-to-wales-78329280

Top Sights in Wales
https://www.lonelyplanet.com/wales

Rail travel in Britain can be expensive. Also, it limits where you can go. Further, once you arrive in a city, you will still have to pay for taxis, etc.

You didn't mention specific places that you wanted to visit.

For a 10 day trip, I recommend narrowing your visit to England and Wales or England and Scotland.

My wife and I had been to London before, so on our recent 28 day driving tour, we skipped London and the SE of England. That is where the worst traffic on the island is.

The Cotswolds are great, as are York, Durham and the Lake District in the North of England.

Wales is very scenic, either north or south. We did the south this time visiting castles and abbeys.
Here is my detailed trip report.
28 days in Britain and Celebrity Eclipse home
https://www.cruisecritic.com/memberreviews/memberreview.cfm?EntryID=599139

We’ve done multiple 2 weeks trips to the UK and have yet to see everything we want. I agree that you need narrow things down to get better responses. I always have a rental car so we can stop where we want and when we want, and be able to stop if we see something of interest. There are many places not easily reachable or regularly serviced by public transport. Where you decide to go will help you decide whether to rent a car or take the train.