There are literally hundreds, even thousands of quests to do. It's best to look at a complete and up-to-date database like eq2.ogaming.com.
Here's a few different types:
Heritage - long quest, detailed plot, hard, good reward. Awards status and guild exp, plus a nice item.
Sabatoge - when trying to sabatoge an opposing faction like Qeynos or Freeport. They give you faction with the benefiting city.
Access - some zones are restricted entry until you complete an access quest.
City Task (Writ) - the four guilds in teh city (fighter, scout, mage, priest) want you to do things for them. The quest raises faction with one of them and lowers with another. Get enough faction with one of the guilds and the merchants unlock items for you to buy. Also grants guild exp.
Collection - find trinkets on the ground, complete a set, turn them in for exp and an item.
Language - learn the language of other creatures. Usually started by inspecting an item, then collect more of the same items and examine.
Lore and Legend - dig into the history of a monster race. Awards a book to be placed in your room and other trinkets.
World Event - major quest line, usually ends with a riad level encounter.
Hallmark - racial or class quest.
Armor - successive quest line grants pieces of a matching set of armor. Starts at level 20.