Currently there are two kinds of quests: Forest Trail, and Big Dig.
In Forest Trail event you’ll complete levels on a forest path to reach the treasure in the end. In this event you do not have to start over in case you fail a level, so you can retry levels.
In Big Dig you play levels to get a shovel for the Hatchling. Once a level is completed, you can choose where the Hatchling will dig the sand. The goal is to find all the keys required to open the reward chest. You won’t be left empty handed even if a key is not found from the hole you chose to dig, as you’ll get other rewards like boosters or coins from all of the holes that do not have a key inside them.