Auron: Find different Jeckt Spheres laying about the world. Some of these will give Auron a new Overdrive. You need to find all ten to get Auron's final overdrive.

Kimari: Use the Dragon Sword ability to drain HP and MP from the enemy. If the enemy has a Blue magic skill that Kimari can learn, he will gain it automatically. The only one that is difficult to get is "Sunshine", which you obtain from the Omega Weapon.

Lulu: Rotate the Right Analog-stick clockwise as fast as possible. The number will go up the more you rotate it. The more you level Lulu up, the easier it is to cast multiple spells.

Rikku: Rikku's overdrive combines items to create special attacks and power-ups during battle. You have to choose any two items to combine. Note: Two of the same kind will work.

Tidus: Just keep using them over and over. After a while, he will learn his next Overdrive.
Wakka: You can win new overdrive roulette (slots) types in the Blitzball Ranking and Tournament modes.