You have four class-exclusive skills. When you start the game, you’ll only start with one skill slot but after a few main story quests later, you’ll gain access to all exclusive skills. When advancing your class, these skills can either be upgraded or replaced. You have to keep leveling up your skills by earning Skill EXP. You can get them through various sources such as unlocking memories, Team Trials, Daily League quests, or through relic vouchers that you can earn from the Arena.
Once you have enough Skill EXP stockpiled, you can access your Skill button and press the Level Up button to spend Skill EXP and gold to level up a skill you want. Additionally, you can just press the Quick Level Up button to automatically distribute the Skill EXP to all skills. There’s no use holding on to your skill EXP unless you’re just focusing on leveling up one skill (which is very inefficient by the way) so don’t hesitate to level up your skills whenever you have enough Skill EXP.