Aside from hacking and slashing your way through monsters, quests are a rewarding, alternative way to level up.
Above their heads, NPCs may have icons which indicate the status of a quest.
Quest Available (Left): A quest is available at this NPC. Speak to them to receive the quest.
Quest In-Progress (Center): A quest you have is not yet completed. Once you complete the quest, return to this NPC.
Quest Complete (Right): You have completed a quest. Speak to this NPC to hand it in.
Quest tracking shows a summary of a quest on the right side of the game. It shows what items you have to collect and how many, which NPC you should speak to next, or where to go next. You can enable quest tracking for a quest on the quest window.
Quest Boards provide a variety of quests starting from Level 1. The rewards for these quests include Experience, Zulie, and unique equipment only obtainable from these quests.