Before starting with the rules...I'll explain the stages. For example:
Charmander: 1st stage
Charmeleon: 2nd stage
Charizard: 3rd stage
And now:
To evolve and grow more power, pokémons need to battle. Of course, every trainer knows about this fact. Here, on this site each defeated oponent gives you ONE point. UNLESS you DEFEAT a pokémon of a superior level to your own Pokémon, in which case your pokémon earns TWO points. Yay! =D
Now, how many points do pokémon need to pass to the next level? Thats why I had to explain the stages above =D
1st stage Pokémons require 2 points to pass to the next level.
2nd stage Pokémons require 3 points to pass to the next level.
3rd stage Pokémons require 3 points to pass to the next level.
3rd stage Psuedo-Legend Pokémons require 4 points to pass to the next level.
(i.e Tyranitar, Metagross, Salamence, Dragonite, Garchomp)
In the end of each battle, (if you're RPing with a mod who is controlling the NPC char/pokémon), the staff member will tell you how many points your pokémon earned. If not, then its your responsability to check how many points it earned.
NOTE: ONLY THE POKÉMONS WHO ARE ACTIVE IN BATTLE WHEN THE OPONENT'S POKÉMON IS DEFEATED EARN POINTS. Means you can change your active pokémon 13252383774728473 times, but only the pokémon who is active when the oponent faints earns points.
In double battles, both your pokémons earn points for each defeated oponent. The same rules apply, but this time you defeat 2 guys instead of just one, and you have 2 pokémons earning experience instead of one. PARTNER BATTLES ARE COOL HUH? =D
NOTE: There can only be, either ONE vs ONE battles, or TWO vs TWO battles. NO three vs three, or four vs four.
If there are 4 trainers competing, using teams, those trainers can only send ONE pokemon for battle each.
To earn moves, etc, you should check, for example, bulbapedia. To see which moves your pokémon gains, at each level.