Buying train tickets and passes

Go to http://www.nationalrail.co.uk to see schedules and which train operators offer services on each route. (This is the official train schedule website.) When you are ready to buy, it will take you to the train operators site so you can purchase tickets.

Before that, I suggest perusing The Man in Seat 61 website to learn everything there is to know about UK trains.

He suggests using a third party to buy tickets. Most of us on this forum don't. but direct.

If that is the order of the itinerary you may very well find that a Britrail pass is the best option. Price out the journeys for say a month from now, and assess how much you value flexibility of travel times against having bargain basement Advance Fares on fixed trains.
Edinburgh to Glasgow you wouldn't want to waste a Britrail pass day on, just buy a ticket and no cheap Advance Fares other than on two mid evening trains.
Southampton to Edinburgh you don't have to route via London. There is an hourly Southampton to Manchester train. Change at Wolverhampton or Stafford for Edinburgh - normally same platform at either station.
That route can be as fast and cheaper than via London The 0915 from Southampton being a very well priced train- about £70 before railcard discount if reasonably lucky by booking as Southampton to Stafford, and Stafford to Edinburgh.
Glasgow to York I would always say to route via Carlisle and the Settle Carlisle line for scenery. Book that as Glasgow to Carlisle, Carlisle to Leeds and Leeds to York no sooner than 8 weeks before travel, and it has the side benefit of being the cheapest route normally.

We use the LNER app for many weeks travel in the UK each year. We buy the Two For One travel pass on the Rail Card app, and that gives us a third off for every trip we take together ( we are always traveling together).
The LNER app has helped us plan where to go too, since we want to know how long train trips are from one location to the next.
I don’t know if LNER is the best way, but it has worked great for us for lots of travel in the last 3 years in the UK.

Just walking up on the day and buying a ticket would be very expensive. If you know which day and time you will be travelling to each place, consider buying an Advance single ticket. This would be for a specifically timed train. Miss it and you would have to buy another ticket. Get there early and you have to watch trains departing to your destination, but you are unable to board.
Book them on www.nationalrail.co.uk around 12 weeks out for the best prices.
There are also various railcards which cost £35 and give 33% off the cost of rail travel after 0930 weekdays and anytime weekends. Assuming there are two of you, a twotogether one would be the best to buy. Still £35 but that's for the the two of you. You only have to spend more than £105 for it to be viable.
You don't need it at the time of booking, just when travelling. They can be ordered online, at a station or my preference, via the railcard app, with it stored digitally on your phone. You just need a couple of passport type photos in your phone's gallery.
Another way of saving money is split ticketing. This is where say you want to travel from A to C on a train that also stops at B on the way. Again you book single tickets, one from A to B and another from B to C. Same train, same seats and the overall journey can be cheaper. Needs careful planning though.

If you do your research on scotrail.co.uk, Scotrail will check to see whether splitting the ticket will save money. It only shows a maximum of one split for each trip; occasionally a second split would save additional money, but you'd have to check for that manually.

First, I'd like to thank everyone who replied to my question about train tickets. I had no idea there were so many options! However, it looks like I can handle it from the US if I buy the a well thought out collection of a tourist passes available to me before I leave home. The idea of having it all taken care of before we get on the boat to Southampton means we can relax and enjoy the ride without thinking about it. Again, thanks for the advice.

One thing to bear in mind is that although there are many operators running different regions and routes, they sell each other's tickets. So that you could use say, Northern Rail to but tickets for another part of the country. The nationalrail website will show all the options.

Have you explored the impact that a two together railcard would make for your journeys? You can buy the tickets at that rate without a railcard, just simply purchase the railcard prior to your first journey.

I also will be in England in September. Yesterday, I downloaded the LNER app and played with it some. You will receive a 5 GBP credit. It appears to be simple to use. It is my understanding the this app has all the British trains on it. You absolutely want to take a look at the two together card. Also, it seems to me that round trip tickets are much less expensive than one-way.

Also if you register (free) for the LNER Perks programme you get 2% off all journeys on LNER trains (only) and 10% (I think) off on board food and beverages ordered to your seat on the app.
I'm not quite sure what happened but there was some problem with perks. Some people, including me, have been given 40% off LNER tickets for a limited period in compensation for who knows what. AFAIK that particular offer is just for old customers. But it just could be for all Club members.
Avanti trains on the West Coast have a similar scheme.