This part will cover the vanilla part, level 1 to 58. You could also do this to level 60, but you can enter Outland at 58, and that is much faster than to level to 60 in the vanilla zones.
You should equip your character with as many pieces of heirloom gear as possible. These are sold in all major cities, and each piece is between 500 and 700 gold, so it's not too expensive. The most vital parts are head, back, chest, pants and weapon. Trinkets are nice, but optional. Rings are pretty difficult to get, but use it if you have it. Amulets are also good to have, but not needed, since they don't give + % XP like the most other parts do. The weapons does not give +XP either, but are essential for leveling fast, since you will be a whole lot stronger with one of these equipped.
Check out all parts of the leveling guide:
Part 1) Level 1 - 58 / vanilla leveling guide.
Part 2) Level 58 - 70 / The Burning Crusade leveling guide.
Part 3) Level 70 - 80 / Wrath of the Lich King leveling guide.
Part 4) Level 80 - 85 / Cataclysm leveling guide.
Part 5) Level 85 - 90 / Mists of Pandaria leveling guide.
Part 6) Level 90 - 100 / Warlords of Draenor leveling guide.
Part 7) Level 100 - 100 / Legion leveling guide.
Level 1 - 15The first part will depend on what race you play as. Some races such as worgen or pandaren has a long intro quest, and you pretty much just need to follow this to be released into the world. For this, just rush trough the quest chain as fast as possible to get this done.
If you play a class that can do dots, such as priest or warlock, run around and dog many mobs. They will die before you need to renew your dots, so you can powerlevel like crazy by just running around and dotting enemies. Combine this with quests to reach level 15 fast.
Non-dotting classes should either just run along with the quests, or go kill mobs a few levels higher. With a full heirloom set, you can easily kill mobs 4-5 levels above you at lower levels, so just rush along and kill a few level 5 enemies after creating the character.
Getting to level 15 should take about one hour with this method, but again, if farming mobs are too boring, just go along with the quests. It won't take much longer.
Level 15 - 58Leveling from level 15 to 58 is fastest done by chaining dungeons. As a tank or healer you will get instant queues, so pick one of these speccs if your class has it. As a DPS, you will mostly get queue times between 2 and 8 minutes, depending on at what times you play.
You should run each dungeon at least once, but queue for a random dungeon, and you will usually get each one at least one time. Make sure to do all the dungeon quests, because these gives an insane amount of experience.
After each run you should invite everyone in the group to join together for more runs, because this will let you skip queue times.
With a group of all heirloom geared people running these dungeons should be very easy, and you can chainpull most of the instance.
Running dungeons all the way to 58 can be somewhat boring for people, and if you need a break, open your Adventure Guide to pick up quests for your level. This can be a nice break, but will take a lot longer than running dungeons.
Once you hit 58, go our to the next part of the guide to level up during the Outland content.