I'm just starting WoW for the first time...

My question: Is the storyline and quests that I'm doing the same as what players did when the game started (in 2002 or whatever)?

Or have events happened in the game that have progressed the entire timeline/story of the game for new players?
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Each continent roughly corresponds to an expansion e.g. an arc in the timeline.

* Outlands: Burning Crusade (2007). It is unmodified.
* Northrend: Wrath of the Lich King (2008). It is unmodified.
* Eastern Kingdoms and Kalimdor: Cataclysm (2010). This expansion completely replaces the quests and map from the classic version of the game. The classic story arc is no longer available on the main servers. Thrall is no longer warchief, Garrosh is.
* Pandaria: Mists of Pandaria (2012). It is unmodified except for the Vale of Eternal Blossoms, which was destroyed. You can't go back to the original Vale. Also, the legendary cape questline is removed from the game.
* Draenor: Warlords of Draenor (2014). The legendary ring questline is removed from the game.
* Broken Isles: Legion (2016). Weapon artifacts still have stats and can be leveled up, but they no longer have abilities. Legendary items, besides your weapon artifact, are disabled.
* Kul Tiras and Zandalar: Battle for Azeroth (2018). Azerite armor, the Heart of Azeroth, the legendary cloak all seem to still be in the game - expect none of these to be useful in shadowlands. Corrupted gear is no longer attainable and corruption effects are deactivated.

You can play in all of the above content to get to level 50. You can start any or all of the expansions, switch expansions at any time, switch back. Whatever you want to do.

For levels 50-60, you'll play in the new Shadowlands (2021) content.

There is one more bit of content to mention:

* Original Eastern Kingdoms and original Kalimdor: Classic (aka Vanilla) (2004). There is a separate installer/server for this content. You can create a separate character to play this content.
The whole world has changed into a cataclysmic world. It should be the same but with more quests and a higher level cap.