Quests
You'll be doing a lot of quests to level up, earn money, and get rewards. Most quests will start out as pretty mundane (go and kill/collect x things), but they will get better. Quest givers will have a yellow exclamation point above their heads.
Right clicking on them will open up their quest box with the details of the quest in it. It will tell you what they want you to do and why. At the bottom you can either accept or decline the quest.
Once you've completed a quest, the yellow exclamation point will become a yellow question mark and you can turn the quest in.
Doing quests will grant you experience to level up, give a little bit of money, and sometimes new gear and weapons.
When you kill a creature in WoW (except for things identified as critters), you will be able to loot its corpse. Just right click on the body and the loot window will pop up. Thats pretty self explanatory.
You can gain experience and sometimes money and loot from mobs that you kill such as animals and humanoids. A creature with its name in yellow is neutral and will not attack you unless you attack it first. A creature with its name in red is hostile and will attack you instantly if you get too close to it.
Neutral
Hostile
When you click on a mob you will be able to see its health bar and mana (if it has one) bar. You will also see its level and the color of the number will indicate the amount of experience you will get and how hard the mob will be to defeat.
A grey number means the mob is way below your level and you will get no experience for killing it.
Green means you will get experience, but it will be very little.
Yellow means you will get a moderate amount of experience.
Orange means the mob is a bit harder to kill but you will get more experience from it.
Red means you will get the max amount of experience possible from that mob but it may be difficult to kill on your own.
If you find a mob with a dragon around its portrait it means one of three things. A golden dragon means the mob is an elite and will be tougher than a normal mob, even if its at your level. All dungeon bosses are elites but you may also find random elites wandering around in the world.
A silver dragon without a wing means the mob is a rare spawn. Rares usually have decent loot and will usually only spawn within a certain time frame. For example, a rare may only appear once every 3 hours in a certain area.
A silver dragon with a wing means the mob is both rare and elite.