You can permantly raise many of your characters' stats through the use of various Sources (ie. Power Sources, Speed Sources). These items are obtained by using the Morph ability on the various monsters found inside the Gelinka (the sunken Shinra airship).

The best place to level up is a specific area inside the North Crater. When you split up your party, if you take the left path, there's another path that branches off at the screen where there appears to be a crack with mako or some substance spouting up from it.

There are enemies inside this area called Magic Pots that give you 1000 AP and a good deal of experience each, but you have to give them Elixirs before you can cause damage to them (they only have 4000 some HP). Stock up on Elixirs by using the W-Item duplication trick, and fight Magic Pots.