For complete spoilers, follow the link to what you're looking for. Most quests open up according to your level. Your progress in most of the quests can be tracked in the Quest Log.
Note that newly-ascended characters can't complete the Toot Oriole's Quest again. They should still visit Mt. Noob for a special delivery.
All other Council quests (except the Strange Leaflet Quest) must be completed before the Naughty Sorceress Quest becomes available.
These quests are given by the NPCs in your Guild Hall, and most only become available when your base mainstat is high enough.
Each stage of the Nemesis quest requires the completion of the previous stage. Similarly, the series of quests given by "Paco" (the quest-giver in the back of the guild) unlock in order.
- This was formerly the level 9 Council quest, until it was replaced on November 15, 2012 and became optional.
Quests That Don't Appear in Your Quest Log
On October 32 (November 1st), 2005, The Council of Loathing gave the Zombie Cure Quest/Doc Galaktik's Quest for a Cure to everyone. The Gray Plague and its quests were ended on November 6th, 2005, after the entire population of the Kingdom was vaccinated. The sick ones, anyway.