In the world of T4C, you can either adventure on your own, killing monsters and accumulating treasure, or you can partake in major and minor quests assigned to you by NPCs. Major quests form the backbone of a large, epic storyline of good versus evil in which you can become an important and heroic player. As you progress, you will find that there is a "Good Quest" and an "Evil Quest." If you go too far into one, you will find it hard to come back and change storylines. As such, you should think carefully before getting involved with the "wrong" kind of people. NPCs remember who you are and what you have done, and may refuse to speak to you altogether if you have become what they consider an unsavory individual.
Minor quests can be done by most people, usually regardless of how "good" or "evil" they are.
Quests can be difficult to find, sometimes. You have to talk to NPCs and focus on their keywords, following their train of thought and figuring out what might interest them. Sometimes, vital keywords they respond to aren't highlighted, so you may want to try some words even if they don't appear important at first. Often, you will also be given choices. Make your decisions carefully. You cannot always come back and change your mind after the fact.
If you want to try your hand at questing, why don't you go visit the Lighthaven Samaritan and mention to him that you'd like to do some errand for him. Upon mentioning that word, he'll ask you to do the Rat Quest (a minor quest asking you to kill plenty of rats).
Quests have a variety of requirements and an even greater variety of rewards. Some of them will give you gold and/or item(s), some others experience points, and some will just ask you to fulfill additional quests. In addition to these rewards, some quests have an "invisible" effect on how NPCs perceive you in general. For instance, if you have performed several good deeds, it may open up some quests while closing some others to you.
The Rat Quest (see just above) rewards you with a few thousand experience points for your trouble.