Each skill (and special) card has a skill level, which goes up to 10. In general, the higher your skill level, the higher the damage and the shorter the cool-down. For skills that do not deal damage (such as buff/debuff), the SP cost (and cool-down time) will go down as you level up.
You can typically farm the 2-star skill cards on event islands and receive one 3-star skill card as a first-time clear bonus for certain event stages. The 4-star version is usually in the coin gacha.
To level up a skill, feed your base card with a card of a same skill. Each compatible card gives 100 skill exp. Different rarity cards can have the same skill, and you can see which cards have the same skill as your base after you have chosen a card to be your base:
Tip: learn the character types and use filter for an easier time
Note that skill exp can be carried over. For example, if you have a 3-star skill card at skill level 5, and later get the 4-star version, you can fuse it into the 4-star and it will become skill level 5.
The skill exp required per level is as follows:
All the above applies to Special skill/attack as well, with different skill exp tnl.