Leveling an arcane mage is a simple process. Most of the talents you want are in the Arcane tree and any branching out will happen at later levels to flesh out the spec. Some players will choose to go with a spec that is a little more survivable, while others will simply level in a raiding spec to get used to the rotations. The best part about playing an arcane mage is you have plenty of options.

Staple talents for the Arcane Mage are:

Arcane Stability 5/5 – Arcane Missiles can not be interrupted
Arcane Concentration 5/5 – Clearcasting for free spells
Presence of Mind 1/1 – Next spell is instant cast
Arcane Potency 2/2 – Increased Crit rate while clearcasting
Arcane Instability 3/3 – Spell Damage and Crit rate increase
Arcane Flows 2/2 – Lowers the cooldown on some major spells
Missile Barrage 5/5 – Speeds up your Arcane Missile Casting

These are baseline talents that every arcane mage will typically have. With these to work around you can decide on many other options.

Some players like to go into the Fire tree for Pyroblast. With this you will lead with the Pyroblast, as it has a long casting time (and does major damage). After Pyroblast, you can cast a Frostbolt to slow your target, or simply start casting arcane missiles. If things get rough you can activate Presence of Mind and cast an instant cast Pyroblast on your target to bring it down quickly.

The one talent in the fire tree that most arcane mages will want to have is Incineration, for the Arcane Blast damage increase. Do not get this talent early, as you will not get that spell until you are higher level.

Frost talents you should consider are Precision (hit rate increase) and Icy Veins (spell haste buff).

There are many options open if you decide to level a mage

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No matter what some people may say, leveling in World of Warcraft can be hard at times. In fact, some classes have it much harder than others. Hybrid classes are able to heal between fights, but often have weakened DPS capabilities to counter that benefit. Straight DPS classes suffer from needing to eat and/or drink between fights to regain health and mana, slowing down the leveling process considerably.

If you find yourself asking how to level fast in World of Warcraft, I can offer a few solutions.

Keep yourself stocked with food/water – While using them causes downtime, it will be much better than trying to regain health and mana without them.
Carry a few potions for emergencies – Potions are fairly common drops in WoW. Keeping a few of them in your backpacks will be handy on close fights. Dieing and having to run back will absolutely kill your experience rate. This tip should be used at all levels, period. There is no excuse for not having at least a stack of potions on hand.
Learn all the fight paths – Moving from location to location takes enough time without having to constantly run between spots. It is better to spend a few extra minutes running to the nearest fight path before questing in an area, than to find out you forgot to after you hearth back to Ironforge.
Keep First Aid capped – Every character needs first aid, even healers. Bandages will easily cut down on your downtime. Food should only be used if your first aid cooldown is not up for maximum experience gain.

You should also consider purchasing a World of Warcraft Leveling Guide Zygor Guides answer the question of how to level fast in World of Warcraft