Welcome to Azeroth! 
PS up front: I see you have fallen into the “New Player” trap. 
It is a BAD IDEA to make your character(s) on realms labelled “New Player”. It used not to be, but the dynamics of realms have changed.
I’m going to give you a short prescription:
- Make Alliance characters on Silvermoon realm
- Make Horde characters on Draenor realm
- If you are interested in Role-Playing, choose Argent Dawn realm
The reasoning behind that would take a book to explain, but I doubt many would disagree with me. Unless you have some specific goal, like performing competitively at world-class level, that is what you should choose. Some might prefer Ravencrest to Silvermoon for Alliance, or Stormscale to Draenor for Horde, but they would allow that there is not a lot between them.
Back to your question.
As I’ve said before, there isn’t a good way to explain this to a new player. Can’t be done. You don’t have the concepts yet.
You can do every quest in every zone, but as Dottie says, there is a ton of story that is simply not in the zones; it’s in the dungeons and raids and end-expansion zones that you don’t use for levelling.
So, during an expansion, people level from, say, 100 to 110. But then there is “end-game”, which is all the content presented for level 110 players during that expansion (Legion). People level from 100 to 110 within two weeks. It can be done in two days. But an expansion lasts two years.
Most of end-game content isn’t about story. It’s repeating the same stuff over and over for increasing rewards. But there is still plenty of story to go round, over two years.
My favourite example of a raid-based story is doing the three dungeons, in order, that lead us to attack the Lich King in the ICC raid.
My favourite example of an end-game based story is the Mists patch story where the Alliance and Horde main fleets land to follow up the initial scouts, and that follows through to the evil plans of the Warchief. (Though I grant Suramar is a much bigger and more complete story.)
The rest is a previous post I made from this thread, which you may find interesting:
I’ve tried to think of the best answer for this question several times.
There is no good answer.
Maybe that’s OK, because WoW is not a linear start-to-finish story. It’s a mesh of different stories from different time, and just one part of a far longer universe (or universes.) 
The best answer I have, if you have a 120, is this:
Choosing: Make 1 Horde character and one Alliance character. One way to choose that might be interesting is to make one character of each race and level them to 5 or so, out of their starting zone. That doesn’t take long, and lets you get a short feel of each character.
Classic: Quest through Kalimdor on your Horde character. Quest through Eastern Kingdoms on your Alliance character. You can do it the other way, but that’s the most natural. Except, BE SURE to run your Horde character through Tirisfal Glades, regardless. It gives a key insight into the Forsaken. Oh, and level your Worgen until you get out of Gilneas. These two zones will explain the Greymane-Sylvanas relationship.
There isn’t much end-game at 60, so you might as well proceed. By all means, see the Classic dungeons and raids, but I wouldn’t say order was important.
60-80: Send one character to Outland and the other to Northrend. Quest through accordingly.
There is quite a bit that is interesting in Northrend at end game. You won’t be able to solo anything on your levelling character, but you can see it on your 120. Do the Heroic dungeons and raids in order - Wowpedia and Wowhead will give you the details. Especially, do the three ICC dungeons in order before ICC itself.
Outland isn’t quite as interesting, and the story is not as coherent, but it’s still worth understanding why Magtheridon is below Hellfire Citadel, what Lady Vashj is up to, and to catch up with Gorefiend, whom you met in Shadowmoon, in Illidan’s Black Temple.
80-90: Same goes for Cataclysm and Mists. One character each. In Cataclysm, the raids just cap the stories which are almost entirely in the zones. In Mists, a lot of the story is told after you have finished questing. I note especially the great 5.1 storyline of Garrosh, Anduin, and the Bell. Isle of Thunder has an entire development and disposal of a huge character. Siege of Orgrimmar and the Scenarios is huge story-wise.
90-100: In Draenor, your Alliance and Horde will start in different zones, so that will get you half way through Draenor on each of them. You can safely split Gorgrond, Spires, Nagrand between them. There isn’t much end-game story left in Draenor now, since Blizzard tore out the Legendary, but the opening of Tanaan Jungle is worth doing.