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.
These guides are completely automated and work in-game, so you can see step-by-step instructions as you play. The guides automatically move to a new step as you complete the quests. You are told where to go i.e. the questgivers for accepting and turning in quests along with where and how to complete the quests. Zygor has more steps and details, which would be more suited to beginners. Dugi condenses the info into fewer steps, but still has all the necessary questing info. 

For Alliance it's Lok'Modan, Redridge, Darkshore, Felwood, Stranglethorn (both parts), Plaguelands. Somewhere between 3 - 3.5 of these get you to 60 in like a 16 hours /played, if you queue for dungeons at the same time. I recommend dropping everything at 45 and going to Darkshore-Felwood if you didn't do them yet - they are literally the fastest levelling zones in the game right now, maybe only trumped by Silverpine.

If you’d rather read than watch, there are also plenty of resources you can use to acclimate yourself to World of Warcraft, figure out what to do next, and ask questions when you’re feeling stuck. For the basics, we recommend reviewing Wowhead’s “Guide on How to Play World of Warcraft,” which is definitely worth scanning if you’re new to massively multiplayer online games or just want to get a sense of how Blizzard’s approach differs from others you’ve played.
I made some route changes to the Horde levels 43-44 sections.  I have swapped 44 Dustwallow Marsh with 44 Desolace (the entire sections). This allowed me to do Deadmire (at lvl 43 instead of 38) and then go stop at TB to turn in Deadmire + The Black Shield at the same time, then fly quickly to do the Desolace stuff.  Doing 44 Desolace is now mandatory because I think its faster with the new routes. This will also make the level 53 grind much shorter.  I think these were great changes.
I've identified a pesky bug that had me confused for months now. I would get serious slowdowns (from 60fps to as low as 20fps) whenever the dugi waypoint arrow was active and I started moving across the map (especially quickly like flying on mounts/flying masters etc.). I tried everything, and all that could fix it was clicking "Remove all waypoints", so that the arrow turned off. I thought maybe the arrow needed to constantly update locations/routes, and a new game update had perhaps broken that feature a bit.

Prepare the Horde for war by recruiting the ancient empire of Zandalar. In this troll-dominated territory, ancient evil waits to be unleashed on the world as you battle crazed blood-troll worshippers, gargantuan dinosaurs, and titan constructs. Discover Zuldazar, the oldest city in Azeroth; unveil the bleak swamps of Nazmir; and traverse the deadly deserts of Vol'dun.

I have tried it a few times. It's good if you're new to the game. Otherwise not so much. I found I leveled slower by using it, as it's suggesting to take the longest possible route usually. Basically following roads everytime, when you could take an easy shortcut. Especially if the shortcut involves a bit of fall damage, it's out of question. The "go here" spots don't also update if you take a shortcut, and then the waypoint arrow points to wrong direction. But perhaps it has been updated since then.

Most classes in World of Warcraft start at first level, but Death Knights and Demon Hunters are exceptions. They start at level 55 and 98, respectively. This alone can shave a huge chunk of time from your leveling experience. Of course, it only works if you want to play a Death Knight or Demon Hunter, but they’re both solid classics that can dish out damage or tank in a group.
The downside is that War Mode will make you vulnerable to other players, which means you might be killed more often. Whether it’s worthwhile will depend on the zones you’re visiting and how active the other faction is on your server cluster. Heavily populated realms usually mean more players on both sides, and more conflict, but on quieter realms, you may rarely run across an enemy.
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.