Each week, you may try your hand at solving a quest. You may try a week's quest only once that week. You gain 10 XP per failure. Once you have won a quest, you may not try any more quests that week. Terrains do not apply to quest, and there are no XP bonuses for staying in-faction and co-aligned. Contact a moderator (or someone authorized to run quests) to play a quest (see the contact page). Immediately after the quest, the moderator will ask what you want to unlock. Be ready with your answer, it's annoying to wait around after a quest while the player reads the text of a bunch of cards, especially when there are others waiting in line to try quests. In other words, choose your prize from the list before requesting a quest!
But most of all, have fun!
*Note: All quests will be available EVERY week, as long as you haven't already completed it, unless noted otherwise.