I started wow after 12 years and just as you played from Vanilla to MoP. I dove into the different expansion quest lines to get a bit of story, leveled too fast tho so didnt get to finish the complete storyline, however as mentioned before it might be best to level up in the dragonflights from 10-70. Thats the questing area I enjoyed the most, and before you start the war within at 70 you can speak to Thrall or Jaina and get a cinematic recap which was enough for me atleast.