Today the character growth system was revealed in Shonen Jump. The scans are from next week's Shonen Jump so they aren't officialy released yet. Therefore the scans are in low quality for the time being.
The growth system is called "Crystalium System" and closely resembles to the sphere grid from FFX. In battle you will earn CP (Crystalium Points) instead of Experience points which aren't in the game (like FFX). There won't be a level-up system in FFXIII! With the earned CP points you will be able to learn an ability and when you learned it another one will open up.