This is my attempt to describe the entire story as I currently understand it, in chronological order.
Disclaimer: I will be wrong about some things and there are gaps in the story which I don't understand.
Year: 2154
-- (begin 
The World We Lost)
The two global powers on Earth, the UNI and IGC, are at war. Fearing that these two powers will destroy the world, a group of 3,000 people build a spaceship to flee the Earth and find a new home. The ship contains the first Titan, named Anicetus. One of the travelers, Andara, leaves a farewell message to anyone on Earth who may be listening then enters cryosleep with the rest of the crew. Anicetus' A.I. aims towards Alpha Centauri and begins the long journey.
Andara
-- (end The World We Lost)
-- (begin 
Conqueror)
The war continues to rage on Earth. You are a soldier for the UNI. The UNI are predominantly Christian and the IGC are predominantly Muslim but you don't believe in either religion. You think religion was created to manipulate and control people and the world would be better off without it. When you receive Andara's farewell message you begin recording your memories and experiences so they can be sent to Anicetus. During the war, the IGC implements a scorched earth policy, nuking everything. This attack kills most of humanity, including members of both sides. You survive by escaping on the spaceship Andromeda.
The war on Earth
-- (end Conqueror)
-- (begin 
Andromeda)
You float in space, orbiting the Earth aboard the Andromeda. You look out the window at the destroyed landscape. You wish you could find a place free from religion and religious wars. If only you could find this Just City and live in peace. You send your memories to Anicetus, hoping they will be received.
-- (end Andromeda)
During the journey to find a new home, 10 individuals fall into a coma and will not be able to wake from cryosleep. Anicetus' A.I. selects these humans to become Titanborn. The A.I. effectively kills these individuals and augments/rewires them so they will be able to communicate with, and control, future Titans.
Aboard Anicetus
Year: 2264
-- (begin 
Progenitor)
After 110 years, Anicetus reaches the planet Acheron. It wakes the humans from cryosleep so they can see this new world. Waking is hard on them and within hours many succumb to cryoshock, including Andara. Anicetus detaches from the ship holding the surviving humans and begins the terraforming process on Acheron. The current atmosphere above Acheron is so thick the humans immediately lose contact with Anicetus. Decades pass. New generations are born on the ship. The population of surviving humans has quadrupled from the original 3,000. The Cepheus Ring, a giant space station encircling Acheron, is built to accommodate them all. A new religion is created around the Titanborn as they continue to sleep.
Sleeping Titanborn
-- (end Progenitor, track 6. Anagenesis)
-- (begin 
As Embers Turn To Dust)
The humans aboard the Cepheus Ring live their lives, waiting for Anicetus to complete the terraforming process. Yet the atmosphere is so thick, the humans aren't sure if progress is being made or if Anicetus was destroyed upon impact on Acheron's surface. Vanguards are sent to Acheron's surface to find Anicetus and report back. None survive. After growing and living on the Cepheus Ring, these Vanguards can't survive on the planet's surface. Eventually, a crew of augmented Vanguards led by Enyo survive the trip and land on Acheron. They marvel at the planet's beauty and witness their first thunderstorm. As the storm passes, they feel themselves being inexplicably drawn towards a faint signal. It's Anicetus; the terraforming process is still under way. The Vanguards rejoice and begin scouting locations for future settlements.
Enyo meets Anicetus and some sort of connection is made between them.
A short while later, on the Cepheus Ring, Enyo observes her twin sister, Alithea, a Titanborn still in cryosleep. Enyo has no interest in Acheron. From the Cepheus Ring she can analyze the sleeping Titanborn and use this knowledge to push human augmentation to unimagined heights. Immortality is within her reach. Enyo builds a group of followers who will leave behind what it means to be human. When the time comes, the people who choose to live on Acheron will look up at the Cepheus Ring and cower in fear.
-- (end As Embers Turn To Dust)
-- (begin Progenitor, track 7. Planetfall)
Finally, Anicetus announces the completion of the terraforming process. This announcement wakes Amyntas, the first Titanborn. He is now forever linked to Anicetus. Amyntas and the survivors walk on the surface of Acheron for the first time. The survivors praise their new gods, the Titanborn, sleeping in orbit. Amyntas vows to protect this planet.
-- (end Progenitor)
These settlers call themselves earthborn, a reminder of the origin of their bloodline.
-- (begin 
Telesterion)
Enyo leads her personal bodyguards, the Banshees, into the Telesterion room of the Cepheus Ring where some of the banner lords have gathered. Enyo believes humanity must augment themselves to survive. She believes humanity is still clinging to their past, unable to move forward. Using The Allodynia Lance, she undergoes a painful augmentation process to give her the same immortality as the Titanborn. She will force everyone to take this next step and destroy any who resist.
Enyo uses the Allodynia Lance
Enyo's followers build more Titans, and set out to terraform Acheron's neighboring planet, Empyrean. Alithea awakens and establishes a link with her Titan, Daedalus.
Hydrus, terraforming Empyrean
-- (end Telesterion)
Year: (unknown)
Empyrean becomes a large industrial planet. More Titans are built on Empyrean. A second Cepheus Ring is built around Empyrean. Between the rings, adjoining Acheron to Empyrean, Daedalus builds the Erebus Bridge. The Erebus Bridge uses the rotational energy of Acheron to generate power for Empyrean.
Acheron, Empyrean, Cepheus Rings, Erebus Bridge
Year: (unknown)
-- (begin 
Siege)
Enyo is done coddling those living on Acheron. She believes they are standing in the way of humanity reaching their true potential. Enyo, her followers in Cepheus, and those on Empyrean, declare war on Acheron. They believe the only possible solution is the genocide of all inhabitants on Acheron. In their mind, "to coexist is to surrender".
-- (begin 
To Coexist Is To Surrender)
Empyrean forces attacks Anicetus, which has become the capitol city of Acheron. The Titan Hydrus leads the attack. A small portion of lucky Acheron citizens escape Acheron aboard specters, their transport ships. Two of the soldiers defending Acheron are Amyntas and Alithea. They are both Titanborn. During the attack, the Titan Virton implements the Empyrean Extermination Sequence, which will destroy all life on Acheron and render the planet uninhabitable.
  Acheron Soldier |
  Empyrean Soldier |
-- (end To Coexist Is To Surrender)
-- (end Siege)
Amyntas is badly damaged during the attack but Alithea escapes aboard her spaceship, the Nefeli. The other survivors who escaped Acheron aboard the specters seek refuge within the Cepheus Ring around Acheron.
-- (begin 
Acheron)
After the Empyrean Extermination Sequence, Empyrean forces land on Acheron to mop up any survivors. Three Empyrean soldiers come across Amyntas, barely alive. They receive orders from the Titan Cepheon not to kill Amyntas, but to wipe his memory instead. Later, Alithea rescues Amyntas and takes him aboard the Nefeli. Looking out over the destruction, Alithea mourns the loss of her planet and vows revenge on Empyrean. They then go to the Cepheus Ring around Acheron to join the other survivors. With no memories of his own and near-fatal wounds, Alithea gives Amyntas the memories of The Soldier of Earth which Anicetus has kept all these years. She then puts him into cryosleep to heal his wounds and soon enters cryosleep herself. They sleep for 400 years. While they sleep, Empyrean forces lock everyone inside the Cepheus Ring.
Acheron burns while Alithea mourns the loss from the Nefeli
-- (end Acheron)
For some reason (I don't know the cause) all of the Titans go dormant after the Siege of Anicetus. Nothing seems to wake them.
Year: 2632
-- (begin 
Empyrean)
During cryosleep, you experience the events of Conqueror and Andromeda as if they were your own memories. You wake up on Empyrean (I don't know how you got there). You have no memories except those of Earth. You don't even know your name; you are just Sentient #2154. After the utter destruction you experienced during the religious war on Earth, you marvel at the beauty of Empyrean's cities. Empyrean seems to be free from religion. It may be the Just City you yearned for aboard the Andromeda.
-- (end Empyrean)
-- (begin 
Cepheus)
Out on the Cepheus Ring, Alithea wakes from cryosleep. You don't remember what happened, but she does. She remembers Empyrean's sins. She is going to exact her revenge and she's going to use you to do it. She needs your help to awaken the Titans.
-- (end Cepheus)
-- (begin 
Xenon)
You marvel at the cities on Empyrean, thinking they may be the Just City. Alithea reminds you that the cities are controlled by machines and lack any human element. The inhabitants of Empyrean worship the Titans as gods. After your experience with religious wars on Earth, people worshipping these man-made creations as gods reminds you how religion is used to manipulate and control people. You agree with Alithea, this is not the Just City.
Alithea reestablishes your Titanborn connection to Anicetus. This triggers your memories. You remember. Your name is Amyntas.
-- (end Xenon)
-- (begin Acheron, track 10. Invictus Daedalus)
Alithea awakens her Titan, Daedalus. Daedalus tells Amyntas:
Amyntas, emperor of titans
Defender of the Earthborn
Acheron awaits your return
-- (end Acheron, track 10. Invictus Daedalus)
-- (begin 
Venator)
Alithea and Amyntas begin their work of avenging Acheron. Their first task: destroy Virton. It was Virton who initiated the Empyrean Extermination Sequence and Alithea has not forgotten. Yet Virton is protected by Hydrus, a "neutral" Titan who has aided both sides during the war. Alithea and Amyntas head to the city where Hydrus resides. They meet with Hydrus and enter an ancient ritual which tests Alithea and Amyntas internally/emotionally. They survive this test and win Hydrus' approval. Hydrus agrees to leave Virton unprotected. Amyntas announces to Virton and Enyo that they are coming. Both sides prepare for the conflict.
-- (end Venator)
Year: 2651
-- (begin 
Cenotaph)
Alithea returns to the surface of Acheron, with Virton floating high above the dead city of Anicetus. Enyo's forces, including Titanborn, Titan Guards, and Banshees, hunt for her. They find her and engage in combat. However, this was all a distraction to get Virton's attention.
Up on the Erebus Bridge, Amyntas convinces Alithea's Titan Daedalus to leave the Bridge and attack Virton. Daedalus collides with Virton and knocks him out of the sky. Virton crashing to the ground creates a wall of fire. Hydrus creates a wall of water to protect Alithea. Alithea and Virton battle. After a long and bloody fight, Alithea is victorious. Badly wounded, she calls for Nefeli to pick her up, much like it did for Amyntas all those years ago.
-- (end Cenotaph)
-- (begin 
Blessings Upon The Field Where Blades Will Flood)
Alithea holding Virton's head after decapitating him
Alithea chops off Virton's head and holds it to the sky, taunting Enyo. Legions from Hydrus and Apheon join Alithea and Amyntas to prepare for war.
Hydrus gives Amyntas her sword, as she holds... the half-Titan daughter of Enyo and Amyntas.