I'd say Fate and Trial are the more important ones. I'm sure training will die out, fun and Basic are for when you're bored and wanna do a quest instead of grind.
They're skills you equip and can be obtained via quest or as a drop. You can only have a total of 4 skills activated at a time.
There's a quest given by Wehn in Pokari City who gives you a choice of 5. You can see them by going to K and there's a button that says Ethyr skills. There are common types as well as class specific ones.