The quickest way to level up is to travel through the game's zones, completing all the quests you encounter along the way. Completing quests grants you Experience points, which will slowly fill in your experience bar with blue or purple. When the bar fills, you reach the next level. Alternatively, you could choose to run Dungeons or Battlegrounds, which are not covered in this beginner's guide.
I am currently working full time on my Vanilla WoW leveling guides. Soon there will be a 1-60 speed leveling guide for Alliance, along with in-game guide versions available for my members area. As of right now, there is no members area, but if you would like to show support for my efforts into this project, you can simply donate to me via Patreon if you like (you will get some cool rewards in return). Or you can donate directly with PayPal if you like. All donations are highly appreciated and they will only motivate me more to continue making the best vanilla WoW leveling guides on the web.
To me, certain character limits tend to be problematic not because of the length of actual content, but because of formatting tags and embedding taking up a lot of characters. This is something I experience on most of websites, not just this one, but the most aggravating issue over here is linking to user profiiles and other sources of strategies, which could be alleviated with internalisation of link paths and @mentions.
Players should stick as close to the level range that we provide below to make sure to maximize the amount of XP they gain. We detail location when it start to get tough to level up after about level 20, this will make it easier to get levels 1-60 with the Warrior class. List of each Location and the Level Range is to the left in a Blue Color for Alliance [Lvl 35-39] and the Orange Color for Horde [Lvl 35-39]. The Mobs you are looking for in that specific area names are Bolded [Mob Name].
Some quests in my guide are marked as "SKIP" and colored in red. These quests are simply either too hard to solo or not worth the XP/time and are skipped. My guide will only list SKIPPED quests if the quest is a direct follow up after completing a quest, not one that you have to click the NPC again to get it. If you hover over the skipped quests, it will give info on why it is skipped in the guide (unless that info is already listed directly in the guide text).
The simple truth behind all of this bickering and squabbling is that there really isn't a bad way to enjoy a game. You like the feeling of Classic taking you back to your young(er) days and that rush of falling in love with WoW for the first time? You like the slower pace of both combat and leveling? You like that the social aspects of the game take a front seat? Good, enjoy it! You prefer modern WoW and BfA with it's tons of improvements, smoother design and more player-friendly features? Great, enjoy it! It doesn't matter whether you prefer modern WoW or Classic or WotLK or whichever your personal high-point was (it was Burning Crusade and Legion for me), you're not wrong. You literally can't be. No matter what anyone says in all these heated discussions, they care about WoW, in whatever form they prefer it. So why can't this be enough of a commonality for us to have a civil discourse?