You can get a 16-25 railcard online or at any station. It costs 24 pound and lasts a year.
East Coast is one of the websites you can buy tickets from. It is always cheaper to buy tickets at least a week in advance online. Don't bother signing up to the national rail website as you can't buy tickets on there. What I love about East Coast is that you can choose your ticket by price. Each train is given a set price between two distances and it varies depending on the destination and number of stops. The more stops, the cheaper it is (takes longer).
http://img191.imageshack.us/img191/7315/77881970.jpg
As you can see for a journey between Newcastle and York, you can book in advance and pay 4.65 or buy on the day for 26.40 for a single.
Splitting tickets works most of the time but it depends when you travel. Basically you find out all the stops you must make on a journey then work out if it will be cheaper to stop at extra or different stations. For example, Stoke on Trent to Newcastle you can go three ways:
1. you change at Manchester. You can choose to get a direct train to Newcastle from there (more expensive) or go to Leeds, Doncaster or York and change again for Newcastle.
2. Change at Crewe and get a train direct to Newcastle
3. Change at Birmingham and get a direct train.
So you could do:
Stoke on trent > Manchester > Newcastle
Stoke on trent > Manchester > Leeds/Doncaster/York > Newcastle
Stoke on trent > Manchester > Doncaster > York > Newcastle
Stoke on trent > Manchester > Leeds > York > Newcastle
Stoke on trent > Crewe> Newcastle
Stoke on Trent > Birmingham > Newcastle
It often works out cheaper to get individual returns (e.g. Stoke to Manchester return, Manchester to York return) or just buy single tickets. Once you find out how many stops you have to take, you can play around and find out the cheapest route. Different operators charge different prices. Virgin are always a few quid cheaper than the rest.
Travelling off peak (after 9.30am) will be cheaper too.
This applies for everyone, regardless of if you have a railcard or not. I highly recommend railcards as you save a third on all journeys and there's one for almost everyone (family, 16-24/young persons, disabled, forces family, senior)