Let's clarify the questlines you can play that came as the "vanilla" version of the original game:

1. Main storyline: a prophet will appear and speak to you in zone 1 of your character's faction. He tells you to come to a place called the harborage. The main story progresses as your character levels. Each ten levels opens a new chapter, leading your character to a confrontation with Molag Bol.

2. Main faction quests: this can be quite confusing now with One Tamriel because your character is off the "rail" and can pick up segments of these all over. Each faction has a quest line that starts in the zone one main city (vukel guard, daggerfall, davons watch). These lines progress through all five zones in that faction. They're quite long. For chronological reference, the end of each faction quest occurs at the point in the main story where you are sent back to Coldharbour at player level 50.

Here's the most confusing thing about the faction quests - you can pick up each zone's first quest even if you haven't finished the previous zones. This can make the story a jumbled mess. If you want to play the story, be careful and don't accidentally pick up the quests in the entrance to a new zone. Online guides can be helpful in knowing what order to play the quests. But in general, the faction quests always direct you forward...they always complete a segment then have a new quest giver right there telling you where to go next.

You may play all three faction storylines simultaneously...they don't really overlap.

3. Mage and fighters guild: these quest lines open as you level these lines. Mage guild levels by picking up purple lore books around Tamriel. Fighters guild levels by killing undead and daedra, and closing dolmens. Each guild has six main quests, which end at the same place the faction quests end, right before you go to Colharbor


Expansion/DLc/other quest lines

Craglorn: several quest lines revolving around a central story, all self contained in craglorn and occurring chronologically after the vanilla game.

Wrothgar: a main storyline and dozens of side quests, all self contained. Occur chronologically after craglorn...or at the same time...doesn't really overlap.

Imperial city: it's a pvp zone, but there's a storyline of pve quests.

Thieve's Guild / hews Bane: main storyline, self contained in zone.

Dark brotherhood / Gold Coast: main story, self contained in zone.

Morrowind: chronologically it comes at the end. However, ESO has been designed to play in whatever order you want (mostly). The worst thing that happens if you play Morrowind before, say, Wrothgar is that at the end of Wrothgar someone appears and says something about Sothol Sil and trouble in Vvardenfell. It won't ruin anything.