There are 5 different skill types in Fernz Gate: Magic, Magic Unisons, Physical Skills, Brave Strike, and Shared Skills. All skills except Brave Strike have a "Cool Time" attached. The "Cool Time" is the number of turns it takes for that skill to cool down before you can use it again.
Magic is broken down into Fire, Water, and Earth, and can be learned by anybody as long as you've equipped the right ring. (Fire Ring lets you learn Fire Magic, Water Ring for Water Magic, and Earth Ring for Earth Magic.) Fire Ring B lets you learn Fire Magic up to level 50, while Fire Ring a lets you learn Fire Magic up to level 99.
Once you've learned a Fire spell (or Water, or Earth), you must keep the appropriate ring equipped in order to use the spell. So, if you equip Fire Ring B and learn the spell "Fire," then you can use "Fire" as long as you keep a Fire Ring equipped. If you remove Fire Ring B, you'll still know "Fire," but you won't be able to use it anymore.
Alternatively, there's also an Elemental Ring B, which lets you learn (and use) all magic up to level 50, and an Elemental Ring a, which lets you learn and use all magic up to level 99.
Once you have mastered all three classes, you can then equip a Reveal Ring, which allows you to use all learned magic. (Note: The Reveal Ring lets you use all learned magic, but it won't let you learn any new magic.)
Magic spells also level up with repeated use, becoming more powerful the more times you use them.
Magic lists can be found here.
Magic Unisons are spells that combine two spells into one powerful spell. (The Unison formula is "Spell A" plus "Spell B" equals "Spell C.") For example, Cure + Cure = "Hi-Cure," which is a stronger Cure skill. Some Unisons use the same spell, and some combine two different spells, such as the strongest Unison, Yggdrasil, which combines Hyperion and Well of Urd.
Unisons can only be used by members of the same team. In order to use a Unison, you need to have one Team member choose "Spell A" and the other Team member choose "Spell B." So, in order to use a Unison, one Team member needs to learn "Spell A" and the other needs to learn "Spell B."
The Unisons list can be found here.
Physical Skills and Brave Strike
Each party member learns their own physical skills, which, like magic, level up with repeated use. Also, each party member (including buddies) has a unique "Brave Strike" which is essentially their limit break. Brave Strikes need to charge up over time, and can only be used when the gauge is full.
A list of all skills and brave strikes can be found here.
Shared skills are skills that you learn from NPCs throughout the course of the game, that can be used by any party member or buddy at any time. Some shared skills are slightly more powerful if used by a specific party member (and those skills will say which party member to use.) All shared skills have a "Cool Time" of 20 turns, so you'll want to use them wisely.
The list of shared skills can be found here.