Skills are used to boost many different parts of the game, from production values to ad boosts.
Each individual continent from the first (Grass) to the sixth (Ancient) will give you the currency needed to purchase Skills for each of the skill branches! (The Lost Desert Continent does not have its own skills in the Research island.)
You can find the Skills in the Research Island on your world map. You can buy Research Points through a number of ways, with the currency used for that branch (example: Ice Cash for Ice Research Points, or Dawn Cash for Dawn Research Points).
You can also purchase Research Points using Super Cash or some offers in the in-game shop!