If you plan on doing much train travel within the UK, do these two things first:

1. Download the Railcard app and buy a Railcard

2. Download the Trainline app to book your tickets

We opted for the Two Together Railcard and it paid for itself with one transaction. It cost £30 and it saved us money on our train to Edinburgh, trains to and from London, and trains to the airport.

We also found that the Trainline app was much better for booking tickets than their main website. The user experience was better and it was easier to find the best deals. When booking your tickets, you just indicate that you have a Railcard and it automatically gives you the best price. If you know you'll be taking the train while visiting the UK, both of these are really easy to do in advance. And booking train tickets in advance is always a good idea. The price you pay day-of from a machine at the station is basically guaranteed to cost more. Also handy: since we weren't staying in London, it was easy to select a train ticket with a Travelcard option to cover all of our transit while in the city. 

Happy travels!