Proper "cutscenes" that completely remove you from the action are fairly rare in WoW; they're typically tied to significant story moments, and there are probably no more than a few dozen per expansion (and they're more common in the newer expansions). Instead, the story and world tends to be communicated through quest dialogue and NPC actions.
If you didn't know, there was also a huge leveling revamp that happened just a few weeks ago. Leveling is now much quicker. New characters will go through a tutorial island called Exile's Reach, then embark on an adventure through the latest expansion (Battle for Azeroth) to reach level 50. Once you're level 50, you can go through Shadowlands (the expansion coming out today!) to level up to the max level cap of 60.
And once you have a level 50 character, subsequent characters that you make can now opt to level through one of the other old expansions instead of BFA. So you can level up entirely by playing through the story of Legion, or Warlords of Draenor, or even Burning Crusade.
Hopefully that answered your question a bit. :) To go back to your original topic: there are no "wrong" ways to play World of Warcraft. Everyone does it a little differently, and I've met a lot of people over the years who just make new characters, experience the world, discover quests, and collect things, all mostly solo. There's definitely no shortage of content if you choose to play that way, and it actually might be better than ever since the new leveling system lets each character delve deep into one expansion story.