There are several ways one can progress: Learning - Training - Applying
Quests will allow you to apply what you've learnt. It's also an opportunity to earn money and even find objects or rarer occassions.
However, if you're too greedy, or careless, quests are also a means of suicide. For when you quest, most of the time you'll be in ennemy territory, or facing ennemy's...
You can start accomplishing quests as of Apprentice - Level 5.
To start a quest, go to the Quest Hall where you'll be assigned your task.
Most quests will be more combat orientated. Meaning it will be mostly to go somewhere and face your opponent. However, it's not impossible to receive textual or part textual quests. These involve less fighting (or at least stat fighting) and more personal writings