Your current quests are located in the bottom left-hand corner of your Polycraft window.
The main quest is in orange and is always on top of your quest list. It always pertains to buildings you need to build or upgrade to progress in the game.
Mini quests are quests that do not help you progress in Polycraft. A maximum of two mini quests will show at any time, underneath your current main quest. Each time you complete a mini quest, the next one will take its place.
Mini quests are comprised mainly of buying decorations from the Decoration Marketplace and defeating a small number of a certain type of enemy.
The last mini quest requires you to defeat the mighty Chiefling, and once all side quests are completed, only the main quest is left.