Trains - should we buy in advance, and if we do, what site to use

For the UK (so not Republic of Ireland) use the train operator web site, rather than a third party. Just to note that any train operator will sell you tickets for any route. The National Rail site is an official site that has journey planners and will then direct you to the operator to book.

Some routes are short commuter routes and you can but kn the day, others it's best to book in advance otherwise it will be very expensive.

You can always buy tickets on the day. However buying in advance (and it is sometimes sufficient to buy just the day before) can save you quite a bit of money.

Use nationalrail.co.uk to look up times and prices. There you will see what the different fares are, and can decide for your self what balance to strike between flexibility and cost.

enter link description herehttps://www.scotrail.co.uk/inspiration-hub/great-scenic-rail-journeys

Scotrail (see above link) is the main operator of trains in Scotland. However, other companies operate between Scotland & England such as LNER, Trans Pennine Express, Lumo & Avanti West Coast. www.nationalrail.co.uk will cover all the above but beware, on some routes, it can be cheaper to split the fare at calling point(s). None of the above sites will tell you where these are.

You should also consider getting a Railcard. https://www.nationalrail.co.uk/railcards/
People may be able to offer you more tips if you give some idea of where you plan to go & number of people.

For the UK (so not Republic of Ireland) use the train operator web site, rather than a third party

Exactly the same thing applies to the Republic of Ireland- use the Irish Rail website.

Depending on your itinerary (especially in Scotland) a Rover Ticket may be useful- https://www.railrover.org/

If you will be using Scotrail, it would be best at this time to use their website. They are using a temporary schedule due to staffing issues and suggest watching for last minute changes.

In Ireland, also consider using coach travel. Irish Rail doesn't go everyhwere and coaches are sometimes a better choice.