Edinburgh,Inverness, Glasgow Itinerary

You probably mean Britrail rather than Eurail.
That is unlikely to be best value.
If you book ahead, on advance fares for specific trains you should get London to Edinburgh for about £50, London to Inverness or Glasgow for around £13, Glasgow to York for around £32 and York to London around £30- around £140 each in total.
Amersham to London Kings Cross you would pay on your Oyster or by contactless.
That is before the substantial savings you would make by buying a £30 Two Together Railcard

The best way to decide what sights you want to see is to get the relevant guidebooks, which may be available in your local library.

While in Edinburgh I'd recommend you look at Mercat Tours for a tour of the Royal Mile. The Mile can be awfully touristy looking to just wander up and down, but a tour will open up the history of what you're seeing.

Stirling Castle was our favourite site in Scotland and an easy train ride from Edinburgh or Glasgow.

Glasgow was an unexpected surprise. We enjoyed the Kelvingrove Art Gallery and Museum.

Inverness is a bit dull , but just out of town is Culloden and also Clava Cairns. I recommend you do your homework before you get there or the significance of both sites will be lost on you.

Edinburgh gets a lot more expensive and massively crowded during the festivals in August. It sounds as if you're planning to start in Edinburgh, which will allow you to avoid that peak period. If the thought of massive cultural festivals excites you, you might want to shuffle your itinerary. In that case, look for Edinburgh lodging right away.

Thank you for the suggestions, especially the BritRail pass, I think we’ll also get the senior discount card as well. I think we’ll change our itinerary a bit and do Scotland for a week and then do York on a different weekend. Is it recommended to get a Scottish rail pass along with the BritRail? I see where the BritRail pass includes Scotland, but wasn’t sure for side trips the Scottish one is needed.

If you are in Edinburgh during August, the Voluntary Guides give FREE tours of the Royal Mile.

http://www.edinburghfestivalguides.org

The guides are docents who are trained to give factual historical information. They do not want to talk about Harry Potter sites, or Outlander sites. But they give factual, historical information and do not try to sensationalize the history like many of the commercial walking tour guides do. Sometimes, you wonder if those commercial tour guides are telling the truth, We really enjoyed our tour with them and even though they do not take tips, we tipped them.

Also, we loved being in Edinburgh in August during the Fringe Festival. Sure it's crowded, but it's worth it to see the shows and the Military Tattoo. Just book your lodgings in advance and understand it will be more expensive in August than other months.

As for the sights, what are you interested in? It's best to get Rick's tour book of England and Scotland and see what interests you.
On this forum, I once recommended (and still do) the Literary Pub tour in Edinburgh, which is a drama played out in front of several pubs about who is the best Scottish writer. You walk from pub to pub, get your drink and go outside and watch the scene, then off to the next pub. We loved it! but we know our Scottish writers. but others said there was too much stuff about the Scottish writers they had never heard of (Scott, Burns, Stevenson, Mcall Smith to name the famous ones) and not enough drinking.

In York, in additional to all the main stuff that the tour books will tell you about, we liked visiting Castle Howard, which a bus ride from York. While we were there, they were filming a period drama for Netflix- Bridgerton!

I loved Castle Howard and agree that it's a wonderful place to stop. The whole area of Yorkshire is pretty grand in my opinion. I definitely want to go back on my next trip.