Major spoilers ahead so be warned!
Major things to keep in mind when reading the manga:
1) Worlds do not exist in the same time, nor do they run parallel to one another.
2) There is one unbreakable law in the Tsubasa!verse: That death is irreversible.
3) Time can be cyclical.
There are also three [[StableTimeLoop Stable Time Loops]] happening throughout the story:
1) Clone!Sakura and Clone!Syaoran's life
2) Acid Tokyo / Clow Country
3) The Real Clow Country
The major event before the start of Tsubasa is the death of Yuuko, the Dimension Witch and Clow Reed's lover. Clow Reed wished, just for a moment, for Yuuko to open her eyes. This went against the One Unbreakable Law and Yuuko was essentially "frozen" in place in her own world, as she would immediately die should she step out of it.
Fei Wong Reed, Clow Reed's brother, saw this event and grew jealous of his brother's power (and possibly unrequited love). He sought to outdo his brother by overcoming the One Unbreakable Law. To do so, Clow Reed set a plan in motion:
1) He captured a girl named Sakura who lived in Clow Country. Sakura had the strange ability to "remember" all worlds she visited, and these memories were etched onto her body.
2) He cloned her so that should his plan fail the first time, he could simply reset and try again (because time is cyclical in this case)
3) He specifically chose guardians to make sure Sakura could make her journey through worlds and fulfill certain important Destinies. He chose a clone of Syaoran to serve as a mindless puppet whose only drive was to protect Sakura. He chose Fai (nee Yuui) to protect her by manipulating his twin brother's death and he attempted to get Kurogane via the death of Kurogane's parents. However, Tomoyo was able to step in and save Kurogane, hence why Fei Wong Reed declares him "the witch's pawn."
''THE TIME LOOPS (The important part)''
1) Clone!Syaoran and Clone!Sakura are brought to Yuuko after they "die" at the final battle. Yuuko offers them both a chance to have happiness, with she and Clow Reed paying the price because it was Clow Reed's wish that brought this tragedy upon them. Clone!Syaoran and Clone!Sakura agree and the two are "reborn" in another world as two normal people. They meet each other again as teenagers and get married and later birth a son - named Syaoran.
Clone!Sakura begins to have visions of an upcoming tragedy. She and Clone!Syaoran discuss it, and decide to offer Syaoran a chance to go attempt to change he future. Clone!Syaoran and Clone!Sakura make a deal with Yuuko in order to give Syaoran a chance - they are trapped in a small tube where they can see one another but cannot touch the other.
Syaoran travels to Clow Country, wherein he meets Sakura. Sakura is a day away from doing an important ceremony that her ancestors have been responsible for for centuries. Syaoran notices a death curse on her and gets worried.
Fei Wong Reed confronts Syaoran, and warns him that tomorrow, Sakura would die. Indeed, the death curse hits just as the ceremony is underway. Nadeshiko freezes time, sacrificing her life, and Fei Wong Reed then offers Syaoran a deal - his freedom for Sakura's continued life. Syaoran agrees, and is taken prisoner. Unfortunately, Sakura is kept prisoner as well. Both are cloned in order to further Fei Wong Reed's plans.
Clone!Sakura is set in a false Clow Country of Fei Wong Reed's own making. Clone!Syaoran is also sent there with no memory of who or what he is in order to get close to Clone!Sakura and accompany her on her journey.
''THUS BEGINS THE START OF TSUBASA''
2) Acid Tokyo / Clow Country
Clone!Sakura and company are sent to Acid Tokyo and there, Sakura leaves one of her feathers to protect the water reservoir. This keeps the water safe and pure which leads to the development of Clow Country several hundred years later. Thus, real!Sakura's future is preserved and kept safe and stable.
3) The Real Clow Country is kept in a day-long loop. When the Tsubasa group arrive, their existence starts to warp the reality and distort the loop. This allows them to reach Fei Wong Reed.
''NOTES ON FAI''
Fai has two curses on him that were both set by Fei Wong Reed. The first was that whenever he found someone with more power than him, he would kill them. The second was that if Ashura-ou was not killed by him, he would collapse the world around him.
The intent of the first was to kill Syaoran once he became too powerful to control. However, Clone!Sakura saw this happening so she changed the future by taking the hit herself. Ashura-ou had also known about the second curse so he did everything he could to make Fai kill him to prevent the activation of the second one. The second was to kill off the remaining members, as by that point Sakura would only need one member to bring her back to Fei Wong Reed. This was foiled by Kurogane, who dragged Fai out.
''NOTES ON ASHURA-OU AND TOMOYO-HIME''
As powerful dreamseers, they met in dreams and together saw a future where Fai was not the one who killed Ashura-ou, it was Kurogane, thus the second curse was activated and consequently Kurogane sacrificed an arm to save Fai. Both had hoped this worst case scenario would not happen, but made precautionary plans in case it did. Thus, Tomoyo-hime placed a protection charm on Kurogane, in the guise of a curse where for each person Kurogane kills he loses some of his strength, to ensure fatal wounds would not kill him.
However, just as they had envisioned, all of these chain of events happened. When one of Ashura-ou's sharp icicle struck Kurogane, causing him to collapse on near death, Tomoyo-hime's charm was activated and spared his life. This gives him an opportunity to kill Ashura-ou, and when the dying king assured Fai his companions would find a way to save him from his other curse, he knew so because of the visions.
Likewise, in Nihon, Tomoyo-hime explained the truth of the charm, and her warning that Kurogane would lose strength from killing a person back in the beginning had in fact came true when he killed the king and lost an arm. She also told Fai about her meeting with Ashura-ou, and how the king did everything he could to ensure Fai would be spared the second curse. Tragically, Ashura-ou lost control of his sanity as his magic grew stronger. Though one thing clear is that both dreamseers cared deeply for Fai and Kurogane's happiness.
''THE FINAL PART''
By defeating Fei Wong Reed, the StableTimeLoops were broken. Clone!Syaoran and Clone!Sakura returned to their originals, however Syaoran ''was still the son of the clones''. Meaning Syaoran's existence is a paradox, and would warp reality around him wherever he went. This is why Syaoran continues on his journey with Fai and Kurogane accompanying him so he will not be alone.