After a successful battle, or interesting post, fulfilling a large character arc, or story arc, a storyteller will reward you with experience for A. certain pokemon, or B. Useable on any pokemon.
For instance, if your pidgey defeats another pidgey, that gets you experience for pidgey, if you with your team just discovered who's been stealing pokemon from Pallet Town, that's experience usable to anyone. Experience for story and plot will always amount to more than battle experience, so it may be in your best interest to focus on story and plot.
As far as using experience, you simply PM a story teller, and let us know where you're spending it, then you subtract that amount from your xp reserves. Once we okay it, you're golden.
When spending experience to upgrade a stat, the cost to upgrade it by one, is the current stat's total, minus any discount your pokemon receives for having dots in something. For instance, Gyarados has 5 dots in attack, so when upgrading his attack, you will receive a 40% discount and reduction in cost to increase it's attack.
Mathematically it looks like this:
Gyarados Attack: 50, to upgrade 50 xp cost, but, that discount of 40% makes it: 30, so it would cost you 30 experience to increase Gyarados' attack to 51.