Paraphrased from the in-game Help:
It looks like you don't "learn" skills in this game, rather you "gain" skills by placing Crystals in your weapon slots. (Weapons appear to have up to four slots, depending on the weapon you equip.)
Skill Crystals come in five varieties: Fire, Water, Wind, Light, and Time. Different crystals have different skills, at different levels.
Skills are further broken up into magic skills (INT), and weapon skills (ATK). However, all skills use MP.
Equipping skills
When you equip a crystal onto your weapon, you can automatically use the skills (assuming you have the MP).
Each weapon has a Main Slot, which it looks like will take any type of Crystal as long as your weapon is a high enough level.
Weapons also have Sub Slots. Sub slots will only take certain elements (and again, only if your weapon is a high enough level).
Example
Rick's Stock Muffs have a Main Slot, and one Sub Slot, both at Level 1. The Sub Slot only takes Light crystals, but you notice that Rick's Main Slot already has a Light crystal equipped...
Well, trade that for a Lv. 1 Fire Silt crystal, and then equip the Lv. 1 Lst Glow Slit crystal in his Sub Slot, and voila! Rick can now use Fire and Light:
(Of course, it's probably even more complicated than that - but I'm still in the first town!)
