You might enjoy this Rick Steves tour: Best of England;
https://www.ricksteves.com/tours/england/best-england-tour
It goes to London, the Cotswolds, Wales, plus additional wonderful places, such as Bath and Blenheim Palace, where Winston Churchill was born. We really enjoyed the Lake District and York on this tour.
Two things that you said made me think this tour could be right for you.
First, you said "We would like to avoid renting a car if possible."
Second, you said "It will be our first visit to the United Kingdom."
This tour begins in Bath and ends in London. After your days in London, you could take the train north to Edinburgh for several days on your own. Fly into Heathrow (London) and back home out of Edinburgh.
Or leave Edinburgh for a future trip when you will explore more of Scotland.
I believe this tour would be a great introduction to England and Wales for you.
It's a sampler of many sights.
A big plus is that transportation, hotels, and many meals are included/planned for you. It makes it easy instead of difficult for your first trip.
We took this tour in 2018 and enjoyed it very much.
In Wales, the tour took us to Conwy Castle and Caernarfon Castle. They made the transportation easy for us.
I myself would hesitate to rent a car on my first visit. But that's just me.
Some of our forum members here love driving in England on the left side of the road, with its many roundabouts, but others love taking the trains in England instead--which is what my husband and I usually do.
We've been to England many times, but enjoyed the Rick Steves tour because all we had to do was sit back, relax and enjoy the sights.