You NEVER need to buy older expansions. Just by paying your subscription you have access to every content of the game except by the content of the newest expansion.
You'll be doing the following contents while leveling:
1-60 = Classic
61-75 = The Burning Crusade
75-80 = Wrath of the Lich King
81-85 = Cataclysm
86-90 = Mists of Pandaria
91-100 = Warlords of Draenor
101-110 = Legion
111-120 = Battle for Azeroth (this is the actual one)
You can do any of the contents of the game just by paying the subscription, except by the contents of 111-120, where you need to buy the expansion. As you level on the game, you'll pass by every past content of the game, you'll see the Lich King, as you'll see Deathwing, you'll do a lot of dungeons with 4 more people, as you'll do questing alone. When you reach the level 120, you'll start to do the actual content of the game, where everyone is playing. Reaching level 120 is the requirement in order to start playing where everyone else is playing too; when you reach it, you'll be able to solo old content in order to farm transmogs (weapon and armor appearances), archievements, mounts, pets, etc. You can do this content with your guild too. The main point is that you don't need to think about buying other expansions, just by paying the monthly subscription, you alread own them all, needing just to buy the newest one.