A quest on Gateway to Reality is the campaign or adventure that your character takes a part in. Most of the experience gaining action happens outside the city in quests. Treasure, items, and other valuable artifacts are found on quests.
A quest can be as simple as trying to blaze a new trail through the Eastern Forest, or a quest can be as difficult as storming a troll stronghold in the northern mountains.
Usually it's pretty obvious how to join a particular quest. Sometimes, there's a quest leader of some sort; in that case, you should contact that person. Other quests are more formal, of the "who wants to come along?" variety; in those cases, leave a message to the referee in charge of the quest (or to Referee if there doesn't appear to be one). Still others are organized by particular groups; in that case, you must go through the group leadership, and you may not be able to go at all if you're not a member of the group.
When you are on the quest, handle your actions the same way normally would, with one exception: if there's a referee assigned to your quest, you send your actions to that particular referee instead of Referee.
There are some advantages to being on a quest. Quests are by far the fastest way to earn money and neat magic stuff. A good quest can net each member tens of thousands of gold pieces' worth of cash and fabulous merchandise.
Second, and maybe more importantly, quests are the best ways to build experience and power. Time and good role playing will increase it as well, but an average role player that participates in many quests will rise in power much faster than a perfect role player who doesn't do anything.