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.
In Kul Tiras, the Alliance's initial attempt to gain the nation's assistance ends disastrously when Katherine Proudmoore, Lord Admiral of Kul Tiras and mother to Archmage Jaina Proudmoore, not only refuses the Alliance's request for aide but also orders Jaina arrested and exiled for her role in the death of her father, Daelin Proudmoore, in the aftermath of the Third War. It quickly becomes apparent that Kul Tiras has become mired in corruption and outside threats, so Anduin and Genn Greymane send Alliance Champions to aid the Kul Tirans and earn their trust. Aiding them in this quest are Flynn Fairwind, a "retired" privateer, and Taelia Fordragon, the idealistic daughter of Bolvar Fordragon, former High Lord of Stormwind and current Lich King of the Undead Scourge. The Alliance is successfully able to earn the trust of Kul Tiras by combating the pirates of the traitorous Ashvane Trading Company in Tirisgrade Sounds, the Naga and corrupted agents of Queen Azshara in Stormsong Valley, and the occult forces of the ancient, ghostly warlord Gorak Tul in Drustvar. After these threats are defeated, Katherine is left despondent at her inability to lead her people, and resolves to help the Alliance find Jaina and reconcile with her. The Alliance eventually track Jaina to the prison island of Fate's End, where Gorak Tul, having survived the events of Drustvar, has kidnapped her and is tormenting her with visions of her past failures. The Alliance champions and Katherine destroy Gorak Tul once and for all and rescue Jaina, who is finally able to let go of the hate, anger and obsession with the past that defined her ever since the destruction of Theramore. The Alliance and Proudmoores then unite to defend the capitol city of Boralus from a massive pirate siege, after which Jaina is named Lord Admiral and pledges Kul Tiras' navy to the Alliance.
After completing your island expedition and second foothold, head back out to your third and final zone to quest in. Note that any legendaries you were using to level have become inactive and are merely stat sticks now. Feel free to replace them with quest gear as you obtain it. Before heading out, feel free to send your three champions out on new missions!
Even though you will invariably do a bit of soloing during your time in World of Warcraft, the game is a lot more fun when you have other people to interact with. Whether you’re starting out, or getting back into the swing of things, having helpful mentors feeding you suggestions (or gear, or gold) doesn’t hurt. The /r/wownoob subreddit is a good place to find friends to quest with, as is the main World of Warcraft subreddit (and don’t forget its helpful “Murloc Monday” thread for newbie Q&A).
Tiragarde Sound and Stormsong Valley are both fairly similar zones, but Stormsong is a little longer, meaning that you might not even have to finish it to hit 120. Tiragarde also comes with a bit of a warning. There are plenty of side quests to do in the zone after you finish the story quests. Don’t do them. It’s not worth the time or the experience you’ll get.