Call of the Primus

On a quite new character, I had to do some of the old campaign quests (2/9) until helping to free the Runecarver's arm. I got The First Move when I turned in The Weak Link and returned to Oribos.
You can talk to Highlord Bolvar Fordragon instead of [npc=177136] to skip until Korthia.

When I skipped it, I immediately got Call of the Primus. Talk to Tal-Inara in Oribos to start the questline for Zereth Mortis.
She also offers a skip, I've you have done the campaign with an alt before. You'll start there with 3/7 chapters of Secrets of the First Ones and unlocked world quests.

Beware that you will miss the rewards of the first three chapters, if you skip them.
Edit: With the Hotfix for March 18th, players who skip the Zereth Mortis story will get the reputation with the Enlightened.

Campaign:
Chapter 1

1. 
   
2. Call of the Primus - Talk to [npc=159478], turn in at [npc=181655] after that
3. 
   
4. A Hasty Voyage - click [npc=182607] behind you. Click [spell=361417] and [spell=361571] to fill the bar. Don't skip the cinematic
   click [spell=368613], [spell=368614] and [spell=368621] to match the emotes.
5. 
   
6. Long Lost Firim - find Firim and click the 4 objects
7. 
   
8. Give Me A Hand - follow [npc=181561], click the door, then [npc=184057] and defeat [npc=184058]
9. 
   
10. A Mutual Exchange - go to south, step on the teleporter and click your extra button [spell=365986]. Kill [npc=184860] and loot the core
    For Research Purposes - collect 12 items
11. 
    
12. The Road to Haven - mount [npc=184082]
13. 
    
14. Forging Connections - go to Haven, talk to [npc=177486]
15. 
    
16. Defending Haven - kill 13 enemies
    Destroying the Destructors - destroy 4 objectives
17. 
    
18. This Old Waystone - follow [npc=178016] to the cave. Stand in front of the waystone to get the extra button [spell=367326]. Click the stone, turn in in Oribos
19. 
    
20. The Forces Gather - return to Zereth Mortis and talk to [npc=177486]
21. 
    

Chapter 1 took about 25 minutes with itemlevel <190.

Chapter 2:

Spoiler

1. 
   
2. Favor of the First Ones - burn 8 objectives
   Necessary Harvest - collect 15 protein from enemies
   Our Forward Scouts - find and talk to 3 [npc=185844]
3. 
   
4. Knowing is Half the Battle - travel to the marked area, meet [npc=181180]
5. 
   
6. Scour the Sands - go to the yellow circles to discover the area. If you lose your stealth, talk to [npc=181180] nearby
   /way 48.43 47.54 1
   /way 51.51 48.59 2
   /way 50.60 45.25 3
   /way 47.99 44.84 4
7. 
   
8. Battle for the Forge - kill 8 enemies
   Together, We Ride - click 5 [npc=181429]
9. 
   
10. In Plain Sight - reach the obelisk and kill enemies behind you until [npc=182357] is attackable. Kill it
11. 19 minutes
    
12. Reinforcements May Be Necessary - return to Haven
13. 
    
14. A Break in Communication - give orders to 3 NPC
    Nothing is True - kill 4 [npc=181884]: throw questitem [item=187839] on [npc=181884]
    This is Your Fault, Fix It - click 5 [npc=181339]
15. 
    
16. Doppelganger Duel - go to the marked area in the middle of the map and fight against this person :-)
17. 
    
18. Fighting for the Forge - talk to [npc=183724]
19. 
    
20. Seeking Haven - click the teleport to return
21. 
    

Chapter 2 took about 40 minutes with itemlevel <200.
I got several itemlevels, about 1900 rep with [faction=2478], some Anima and 12 renown so far (chapter 1 & 2).

Chapter 3:

Spoiler

1. 
   
2. Danger Near and Far - go to the marked area and enter the building
3. 
   
4. A Mysterious Voice - click the door, watch dialogue and kill attacking enemies
5. 
   
6. Core of the Matter - return to Haven and talk to [npc=177486]
7. 
   
8. Seeking the Unknown - enter the building at 33.63 65.78 and click the information
9. 
   
10. Finding Firim - find the NPC, click its corpse and go to the cave
11. 
    
12. Security Measures - click the [object=373576] near you. Leave the cave and reset 10 traps
    Unseen Agents - kill 3 enemies
13. 
    
14. Now You May Speak - talk to [npc=177958] and click the console
15. 
    
16. Surveying Cyphers - examine the 3 concordances marked on your map. Visit the fourth location. Kill [npc=179939] and click the item behind him. Return to the cave
17. 
    
18. Cyphers of the First Ones - click the console again. Unlock the first option and get your Pocopoc
19. 
    
20. The Way Forward - return to Haven
21. 
    

Chapter 3 took about 30 minutes with itemlevel 200. It gave less rewards than the previous ones.
In total I got about 2k reputation, 15 itemlevels and 15 renown.

Accept [quest=65324] before you go on with anything.

if you hit max level with your first character pretty late (like me) this quest won't pop up when hitting 60 or by just logging in (like mentioned in a lot of guides).

you'll have to finish the campaign till Charge of the Covenants - THEN Call of the Primus will pop up.

Attention! Be aware that by using the SKIP option you will miss out on some rewards including some 233 ilvl items and a ton of renown.
May be especially interesting for fresh alts which are low equipped.
For example you will miss out on these quests and their rewards:
(item selection based on a leather class but maybe similar for place/cloth/mail)

Our Forward Scouts ilvl 226 belt (equivalent to an armor token for 500 anima)
Favor of the First Ones ilvl 233 mail feet 12%move and mount speed (mount speed doesn't stack tho)
Necessary Harvest ilvl 233 gloves 25% hp/mana rezz/ankh
Seeking Haven# ilvl 233 leather/plate gloves with blink
Harmony and Discord ilvl 226 shoulder piece (uninteresting since there is a 233 shoulder a few quests after that)
Cyphers of the First Ones ilvl 233 shoulder piece
The Way Forward# ilvl 226 (equivalent to an armor token for 500 anima)

and a few more ...

edit: you also get 2-3 k reputation less by skipping, legendary flux and cyphers