Process for getting train tickets

You can do what you're proposing to do, but it will cost you. UK train tickets bought late can be very expensive--even more so if you buy them on the day of travel. There is considerable money to be saved on many routes by buying earlier.

To see the approximate difference between buying early and buying late, take a look at fares for tomorrow (It's already tomorrow in the UK, so you should be seeing same-day prices) or for a few days from now, then look at fares for the days you really intend to travel (or as far out as www.nationalrailco.uk will let you go if your trip is beyond the date for which tickets have been released). Do you see "Advance" tickets listed? That's the kind of price you can get by buying early. The fares may still vary somewhat, but I think you'll observe a major gap between fares for tomorrow/this week and fare well into the future.

If you're uncertain about your travel dates or whether the trip will even happen, you probably don't want to buy train tickets very early.

You might also consider a Two Together rail card if there are two of you who will always be taking the same trains. The discount (1/3 off) should easily pay for the card (you only need one for two people), especially if you feel you need to wait to buy your tickets until shortly before you travel.

If you wait until you're in England/Scotland to buy your tickets, you can do so on the website (there's probably also an app, but I haven't checked), at a ticket machine or at a staffed counter.

I assume you can have both tickets on the same phone to show the conductor, but that's not something I've ever done.

Thank you very much for the replies. The two together idea is very helpful as there will be two of us always together. I saw a long list of possible types of fares and was confused by this list of 10 or so options.
Two of us are in our sixties and two sons ages 26 and 28 but they won’t both be there all of the time of our trip.
Is the two together the best option for us and whichever is the standard adult (I couldn’t tell which this was) for each of the young adults?

The lads would be eligible for the 26 to 30 Railcard https://www.26-30railcard.co.uk/ They can travel with you or on their own. £12 minimum before 10am M-F to prevent commuters using it.

The two of you would be eligible for a Two Together Railcard if the two of you will be together every time you use it. It is not valid for one on their own at any time. https://www.twotogether-railcard.co.uk/

The two together railcard can be bought for any two people. But those people have their pictures on the railcard and can't change, so there needs to be a commitment for them to both be on all trips with the Railcard.

From Chester to York, the cheapest way is usually Merseyrail to Liverpool Lime Street, then TransPennine Liverpool to York.

You don't buy tickets on National Rail, you buy them on a Train Company website. Any will do, but as 3 of your 4 legs will more than likely be on LNER, their website is probably best, especially as that website works particularly well for Americans.

You will almost always get advance fares, even on the day, so should never ever end up paying full fare, just the advance fares are higher than if purchased several weeks ahead. The only time you will pay full fare is if you are on an early morning service or on a London commuter line.
All the company websites will always give you the cheapest option. Take that, don't sort through the endless list of other fares for the same train.
LNER, like every other train company, has an App.

There has been much talk about fares going up, and being really expensive. Unless you are a commuter that is NOT always factual and up to date. Locally in NW England quite a lot of fares have gone down. One local journey I have to do in a week's time has gone down from £23 to £16, both of which are walk up fares.

The other day I booked an under £15 fare for a 340 mile journey, which actually starts on the busiest train of the day on my local line. That was booked at what I know from experience to be the "sweet spot" on Avanti trains- about 1 month from travel (not the 3 months quoted, even on Man in Seat 61). That is great value for money.

Thank you! So I think we will go to the LNER website and buy all the tickets there. And possibly the two together if they show that as the cheapest option.
I hope I understood correctly.

Railcards should be bought either from a separate website or at a station when you arrive, but not from LNER. if you would prefer an actual pass rather than a virtual one on your phone wait until you get here. You can download and complete the form in advance and bring it with you to speed things up.
However, although you can’t but the railcard from LNER you can buy your tickets there saying that you have the railcard to get the discount. You don’t physically need a railcard until you get on the train.

This is the Two Together Railcard website-
https://www.twotogether-railcard.co.uk/