In Final Fantasy X making your characters stronger is a lot different then in other FF's. You get AP when you defeated enemies and when your have enough AP you go level up. But that will not increase your stats as in earlier FF's. You must do that on the Sphere Grid. 

¤ Choose a Sphere Grid

When you choose to play a new game you can choose or you want the "Standard Sphere Grid" or the "Expert Sphere Grid".

- Standard Sphere Grid: For beginners, this option is recommended by people who plays Final Fantasy X for the first time. 

- Expert Sphere Grid: For experts, this grid have more options, but it's hard to players who are not known with the standard grid. But this one is greater anyway :D

¤ Getting your way on the Sphere Grid

On the Sphere Grid you can activate nods depending on how many levels you currently have. You can activate nods by using Spheres. For example to activate a Strength node, you need a "Power Sphere". Those spheres you get when you win battles. There are also locks, from Lv. 1 to 4. You just need a Lv. 1/4 Key Sphere to turn that node into a empty node. These locks appear close to supreme spells like "Ultima".
When moving you see lines, that's the line you followed, when you need to return you can move 4 nods, but you only spill 1 S. Lv.

¤ Colors

Each character leave its own color behind when he/she activates a node.