Prologue Mission: 
- Zeratul recieves a "distress" call from Templar Talis that Templars from colony worlds were being abducted
-Zeratul goes to a Moebius Corps facility that is creating hybrid
-Kerrigan destroys the facility after Zeratul frees the capture Templar
-Zeratul finds out that the Tal'darim were behind these abductions 
-Zeratul infiltrates the Temple of Erris and encounters Amon's forces. Kinda like that Tomb of Sargeras mission from Warcraft Frozen Throne

Legacy of the Void:
-Zeratul immediately tells Artanis to halt the invasion of Aiur and warns him about the upcoming events
-Artanis does the invasion anyway, and it's pretty bad ass
-During the invasion, the Protoss was meant to be fighting feral Zerg, but the Zerg were more co-ordinated near the end of the mission in which Hybrids started appearing.
-Amon has "inifltrated" the Khala and pretty much went made all Khalai Protoss under his command. Since the Nerazim has severed their Nerve cords, they are not affected. Zeratul had to fight some Protoss to rescue Artanis (who is corrupted)
-In an attempt to save Artanis from Amon's grasp, Zeratul pretty much went on  a 1v1 with Artanis, in which he died, but freeing Artanis in the process (RIP ZERATUL ; - ; )
-Artanis along with the Nerazim, Karax (a Phase-smith, which is like the Protoss Rory Swann) alongside with some of the Protoss warriors that have severed their nerve cords reactiveates the Spear of Adun to get off-world.
-Rohana, a Grand Preserver becomes like an advisor for Artanis. She's pretty much compares and contrasts the present with the past. 
-Artanis goes to Korhal to get the"Keystone" (the Artifact)  from Raynor. However, the Mobius Corps, under the posession of Amon has sieged Korhal and have taken the Keystone. After a long, dark and hard ( ͡° ͜ʖ ͡°) battle, Artanis and friends were able to get the Keystone back. 

This next part can come in any order if memory recalls;

Nerazim: 
-Since the Nerazim is pretty much the last "free" Protoss, Artanis and co goes to Shakuras. It is too late however, since the Zerg have invaded. 
-Artanis unites with Matriarch Vorazun (daughter of Matriarch  Raszagal) and decides that Shakuras should be destroyed instead, rather than be infested by the Zerg
-After luring as many Zerg in, Shakuras does one of the massive Jihads and kills about a Billion? (I can't remember the exact numnber). The cutscene is pretty badass too, Artanis just being the BMF he is. 

The Purifiers:
-To aid Artanis and co, they have decided to re-activate the "Purifiers" 
-They found a robotic clone of Fenix in one mission
-Artanis learns why the Purifier program had to be shutdown
-They get the help from the Purifiers anyway, and it turned out really well. 

Ulna:
-The Keystone leads Artanis and co to the Ulna, which is like this huge planet-sized diamond-shaped structure where the Xel'Naga sleeps
-Amons forces, the Hybrds, Moebius Corps and the Tal'darim werealos there
-Artanis meets Kerrigan and joins forces to defeat Amon's forces while they search for answers.
-The two learned some answeres but found that they were too late, Amon have already killed the Xel'Naga. 
-Amon had opened a portal to the Void in Ulna. At the same time, a significant Tal'darim commander, "Alark"  boards the Spear of Adun and warns Vorazun (who was put in charge while Artanis is gone) that Artanis is in danger. 
-Artanis and co along with Kerrigan seals the portal and does their own thing

Tal'darim:
-After returning from Ulna, Alark and Artanis makes a deal. Since the Tal'darim were betrayed by Amon, he sought revenge. To do this, he will need to be the new Highlord of the Tal'darim. In exchange, Artanis will no longer have to deal agains the Tal'darim, so they made a deal.
-Alark asks Artanis to get his Templars to kill possible Tal'darim that may contest him for the position once he becomes the Highlord.
-In a ritual/tradition battle, Alark was successful and becomes Highlord of the Tal'darim
-Tal'darim and the Daelaam unites

Final missions:
-The unified Protoss had to disable crystals that will prevent the Golden Armada (posessed by Amon) from Warping in to Aiur to defend Amon while he completes his physical form. 
-Amon's physical form is almost complete, the unified Protoss just had to destroy 5 more crystals to dtop him. 
-Amon completes his physical form, but Artanis does one of the  most  massive overkill in which he ordered the Spear of Adun, the Tal'darim Death Fleet and the the Purifers' ship to launch an all-out orbital attack from their ships. It's like nuke it from orbit but focused on one target, that much overkill.
-Amon's physical form is pretty much destroyed, but the Templa are still not free
-Similar to Wings of Liberty's final mission, "All in", the plan is to use the Keystone to clense the corrupted Templar, likje what Jim and friends did with Kerrigan. 
-They did this last stand thing again, but the Golden Armada arrived. The Spear of Adun was getting rekt in orbit, and as you progress through the mission, you slowly lose the ability to use the Spear's support. (Parts of the Spear of Adun crashes down to the battlefield) 
-Eventually, ALL Templar that was ubnder Amon's control were sent in, but the Keystone was activated. 
Cutscene:
-After the Keystone was activated, Artanis convinces Selendis that she must severe her nerv cord. All the other Templars did the same. 
-Artanis talks about rebuilding the Protoss race, fast forward with actual Protoss rebuilding.It's a really neat cutscene O must say. 

Epilogue:
-Kerrigan calls Raynor and Artanis to Ulna, because apaprently Kerrigan is hearing voices, something important
-The first mission, you play as the Protoss. You pretty much have to kill Samir Duran in his Xel'Naga form
-The three commanders (Artanis, Rayor and Kerrigan) meets with the last Xel'Naga alive. They talk about the cycle of life and how Kerrigan absorb the purities and become a Xe'Naga herself, so she does. 
-Second mission plays you as Terran, you have to defend Kerrigan while she absorbs the powers of the last Xel'Naga
-Third Mission plays you as the Zerg. Kerrigan has become a Xel'Naga. She pretty much became like Phoenix from X-Men
-Once again, they have to destroy crytals to defeat Amon's barrier
-Kerrigan wrecks Amon 

2 years later, Raynor is in Joey's bar, where we can hear a news report saying that the Dominion, under Valerian's rule is sooo much better, and they were celebrating the 3rd Anniversary of Arcturs Mengsk's defeat. (It must have been awkward for Valerian lol) 
Kerrigan apaprently appeared by the doorway, in an angle where the light behind her covers her face, and Raynor goes to her. (I'm suspecting that this is a dark ending, despite it being shown as a happy in ending)
-In a slideshow, it is said that Raynor was never heard of again, and he left his badge on the bar. 
-Admiral Horner becomes successful apparently
-Peace between the Protoss and the Terrans if I recall
-Zagara becomes the new queen of the Zerg, she took control of the planets surrounding Char
-For unknown reasons, life sprouted in barren wolrds in the Koprulu Sector
-Alark refused to be assimilated with the new unified Protoss and went looking for new worlds to colonized. He did give an option to the the Tal'darim to go with Artanis if they choose so

Aaaand that's pretty much it