The power of positive thinking
To get a sense for what wielding magic might be like (and why not everyone can do it), consider the act you have taken in accessing this website -- the tools of the computer, mouse, and keyboard are probably readily available to you, and you can use them without much thought. You don't need to think too much about the electricity that powers these tools or the exact process which brought them into being; but if you did understand such complexities, you might be able to go the next step and build a custom computer of your own. On the other hand, if you had lived your whole life without anything like these tools only to encounter them in your adulthood, you might find acquisition of simple computer usage skills so difficult that they are not worth the trouble, no matter how useful computers could potentially be to you.
In the same way, a beginning mage needn't necessarily understand the intricate workings of magical energies in order to learn his or her first magic spells -- he or she simply needs to open the mind to doing something a little different, something that you might never have thought to do if someone hadn't taught you how. A typical person goes about life without ever sensing the presence of magic in the environment and in themselves, just like someone may see an autostereogram every day and yet never know that it contains a special 3D image if you look at it in the right way -- but to one who has mastered the trick of it, it becomes as simple as riding a bike.
Your character may have learned simple magical abilities through talent, luck, and perceptive insights, but the best and most difficult spells only come after many years of training -- unless, like Mozart, your character is a prodigy of some sort. The World of Warcraft being as fantastic as it is, such prodigies ought not to be so extremely rare.
Another crucial thing to remember about arcane magic in Azeroth is that, unlike the divine magic of nature, spirits, and the Light, arcane magic has been tainted through association with the Burning Legion. The Well of Eternity, which brought arcane magic to Azeroth in the first place, was originally a gift of the Titans, but since the "Dark Titan" Sargeras began destroying the universe with his Burning Legion, he and his demonic forces have largely corrupted this form of energy. The evil effect on a person is nowhere near as powerful as that of Fel magic (which is derived from the arcane), but it still has its impact.
The use of magic serves as a kind of beacon to the Burning Legion, and has more than once been the lure which drew the demons to Azeroth and threatened to wipe out all life there. Also, as the history of the blood elves has shown, magic is extremely addictive, especially when a race has used it for many thousands of years. Various people have tried to find the right balance when using magic, but others (most notably the night elves) have decided it's too risky to handle it at all, and have advocated banning it altogether.
Humans are the backbone of magical studies in the Alliance these days. Although they were originally taught magic by the high elves long long ago, the descendants of those elves have, except for the few elves remaining in the Alliance-aligned "Silver Covenant," either been killed by the Scourge or joined the Horde as blood elves. Even all those centuries ago when the elves first taught them the power of arcane magic, many humans displayed a great talent for it.
Humans also founded the city of Dalaran in the Alterac Mountains, which eventually became the de facto magical capital of the world. Many of the most prominent magic-using characters in Warcraft lore had their beginnings there, and it's quite likely that the majority of mages alive today would have spent a good deal of time there at some point in their past. The various battles that have taken place there since the Second War (about 20 years prior to the current setting) might have affected your character a great deal, especially the latest one during the Third War (about 7 years ago), in which the Scourge and the Burning Legion together destroyed the city pretty bad.
From then until now, Dalaran was sealed off from the rest of the world, rebuilding inside a big magical bubble-shield, so no player characters could have interacted with it during that time. Now that it is open again (and floating over Northrend), your character may have a chance to revisit old memories, however. Feel free to walk about Dalaran with your friends and pretend that you recognize various parts of it, and comment upon what all has changed since you were there last.
Two other races of the Alliance are very talented in magic as well, but neither has interacted with the rest of Azeroth until recent times. The gnomes practiced magic a great deal in Gnomeregan, but until that city fell to the troggs, they didn't venture outward in any large numbers -- still, if your character is a gnome, it shouldn't be too unlikely for him or her to have left Gnomeregan and studied with the humans. The draenei, too, have been great practitioners of magic for many eons, ever since their original days on the planet Argus, when they were still known as eredar. Their millennia of flight from the Burning Legion and near-extermination at the hands of the orcs could have somewhat reduced their knowledge and power since then, however; the latest advancements in magical knowledge by other races may be secrets lost to the draenei many many years ago.
The blood elves share a history with the old high elves, of course, so much of their history is the same as that of the humans, with the one big exception that the elves had the great magical kingdom of Quel'thalas, which they claim held secrets that the humans never dreamed of. Since the Scourge corrupted their Sunwell there and nearly wiped them out during the Third War, there has been a great deal of confusion about where the magical energies they needed to survive would come from, but the events of the Burning Crusade seem to have sorted that out for the most part. Still, blood elves remain very serious students of the ways of magic, and have even laid claim to a portion of Dalaran for themselves.
The Forsaken, likewise, share a great deal of history with the humans. The effect of undeath on their cold and lifeless bodies may mean that the power of magic is less corrupting and addictive to them than to other races, although how much that is true for your character is certainly up to you to decide for yourself.
The trolls are a somewhat different case from any of the other races, since they have long practiced a mix of arcane and divine magic in their voodoo religion. Many of the trollish mages in the Horde's Darkspear Tribe may have once been magic users of other sorts (such as Witch Doctors) who refined and modified their abilities once the Forsaken (and later blood elves) joined the Horde and shared with them the magical heritage of the humans and elves. Like the humans, trolls have an amazing talent for magic and pick it up very quickly.
All the World's a Stage continues this series on roleplaying within the lore with today's look at Mages. Be sure to check out the Warlocks, Paladins, and Warriors (Horde and Alliance). For more about roleplaying the different classes, see how spells themselves can be used in roleplaying, for Druids, Hunters, and Mages, as well as Paladins, Priests, Shamans, Warlocks and Warriors.