Goals (sometimes referred to as 'quests' by players) are activities you receive around the island. They usually involve you fulfilling a task and then receiving a reward. The details vary by the goal, but most generally require defeating or catching monsters.
There are however special dialogue-centric goals that require you to interact with characters in the world. Sometimes you’ll be presented with different response choices that slightly alter the conversation. Other times, you’ll need to actively persuade a character to help you by selecting the correct dialogue options from a list of possible responses.
The goals you've received, as well as the rewards and your progress, are described in your Goal Log. You can reach this by tapping the Goals icon on the right side of the screen.
The Goal Log also has helpful tutorials for each goal type if you get stuck, as well as a Go button that will take you straight to the goal’s objective.