I've put some thought into this after reading a chapter on procedurally generating quest lines (which didn't label types itself). I began thinking in out various types quests and sub-quests. And it seems that most can be categorized into a few types. I may have to come back later after I've thought of it again, since it's been a long day and my brain is a bit fried from tiredness.
Some examples:
- Find an Item: Includes fetch quests (bring item to NPC) and resource quests (find item to use)
- Kill an Enemy: This includes the basic "kill quest" but also is often the focus of a main quest (kill the final boss)
- Find Information: This is like a quest for an item to use but without the item; example, find the old wise-woman of the forest who knows know where a special item is or who the villain may be (she gives info, but no item). I count quest to learn a skill here too.
- Find a Location: For example, you have already found some special spell, but it needs to be cast in a certain place -- so find it.
- Escort Quest: Take an NPC somewhere for whatever reason
- Rescue Quest: Like escort, except you rescue them first
- Battle / Defend Quest: Defend a fort, city, or some other location or some VIP NPC from attackers.
- Conquest Quest: Capture / take control of a fort, city, dungeon, etc.
- Delivery Quest: Bring an item or message to an NPC
I'm sure there are variants I haven't thought of and that others might sub-divide them differently, but I think most quest lines in RPGs and RPG like games can be broken down into a series of such step, or else are just one of these for short throw-away quests.
https://www.gamedev.net/forums/topic/700942-types-of-missions-quests/?tab=comments#comment-5401486