By constantly doing quests, you will experience a very different game than if you just level through pure grinding. You'll be out there making a contribution towards the Alliance or the Horde. Doing these quests takes you through the heart of the game's story AND provides the fastest rate of experience. Have fun in World of Warcraft and enjoy developing your hero - your adventures will gain you great reputation amongst the land.
As you're leveling, I generally recommend you just follow the natural flow of the progression through each zone. They're laid out fairly reasonably. But don't be afraid to hop around, especially if you're near a quest objective. You shouldn't actually need ALL of the zones listed to reach 62. Pick the ones you like best. It's much more important to simply be as focused as possible and flow from one quest to the next.
I found that in most cases, I could pick the zones I liked and skip everything else. It was slower, but I still wasn't forced to do quest zones I hated (and, notably, I skipped the entire Cataclysm expansion's content -- sorry, Thrall.) I did do dungeons, both in group finder and by two-boxing (more on that in a moment as well), but only once for the quests, and mostly to clear what felt like an awfully long slog from 40 to level 80.
Edit 20 It’s been a week since this post and a lot of people are still reaching out to me with questions which is fine. I have had a few different people reach out to me to help make YouTube videos based off this guide and more info in BFA which I’m more than willing to do. Whoever gets the videos up first I will link them here for you guys to enjoy.