Download the game right now, create an account (a new one; not your old one, if you were a previous player), and you’ll be able to level your character all the way to 20 for free. There are other annoying limitations placed on this free experience, but it’s still a great way to familiarize yourself with the world, see if you enjoy the experience as much as you were hoping, and get a feel for the different classes before you settle on a main one to play as for the next 100 levels’ worth of gaming.
The best way to collect AP is through world quests, a system that allows players to complete menial tasks to earn moderate rewards. A finite amount of AP spawns on the map through these quests — generally at least 2,000 a day. I’ve likely done most of these quests close to 20 times in the course of preparing my various characters in the 45 days since the expansion launched. In some cases, I’ve replayed them as many as 50 times.
And, of course, there are plenty of in-game addons you can use to make your adventures in Azeroth even more efficient. Try checking out Azeroth Auto Pilot, which points you where to go, handles all your NPC discussions and quest turn-ins, and automatically sells your junk gear (among other features). Deadly Boss Mods is still a must-have, so you don’t die in dungeons quite so quickly. Pawn helps you tell whether the gear you’re getting is actually an upgrade or not. World Quests List gives you a much better way to see what’s happening in your zone (and sort for the rewards you’re prioritizing).
Island expeditions are very easy to complete while leveling, particularly before hitting Level 116 when your legendaries are still in effect and your secondary stats have not been squished by leveling scaling yet. Island expeditions can be completed in 5 to 10 minutes relatively reliably while also acquiring some of the pets, mounts, and Bind-on-Equip gear available. There is also one weekly quest to do one island expedition, so for slowly leveling alternative characters, be sure to complete at least one island expedition each week.
Currently, it is believed that the War Mode rewards will change depending on which faction is more heavily prevalent with War Mode activated. The Horde have typically had more participating in War Mode, so the reward has been increased for the Alliance to incentivize more participation in War Mode. These rewards could potentially be increased to both the Horde and Alliance to incentivize both factions to higher amounts of War Mode participation.
Also note that power leveling is sometimes used to refer to paying someone else to level your character to cap as fast as possible (i.e. they power level for you); a distinction used by Blizzard is to refer to the paid leveling as "power leveling service." This behavior is against Blizzard's Terms of Service. Power leveling yourself, which this article covers, is perfectly acceptable.
No matter your experience level—whether you’re about to experience the thrill of hearing the Stormwind City theme as you walk through its epic gates for the first time, or whether you can spell Ahn’Qiraj off the top of your head—here are some tips and suggestions for making the most of your new life in the World of Warcraft (or your epic, Illidan-like return to Azeroth.)