SpoilerIshgardians dislike the Sharlayan way, finding it hocus pocus nonsense. However your whole questline revolves about proving them wrong.
While there are Ishgardians, like Jannequinard who are actively rooting for the Sharlayan way, only seems to get away with it due to his father being due to being the second son of house Durendaire. His uncle, Forlemont, is the head of the Observatorium in Coerthas, finding Sharlayan magic a bunch of nonsense.

In the meanwhile Jannequinard invites Sharlayan astrologians over to Ishgard, to come teach him and the ones willing; but mostly to show the Ishgardians that the Sharlayan way has its merits as well.
Sadly enough they somehow end up crashing in the Shroud instead of ending up where they were supposed to meet. You go out to investigate.

However at freeing the two Sharlayans from bandits you soon find out that the Forum, the ruling body of Sharlayan, does not approve of this at all, and even going as far to send people after the the Sharlayan astrologer grandfather, Mace, and his granddaughter, Leveva.

After saving the two from mercenaries Mace finding himself too old to teach properly, and leaves you with Jannequinard with his granddaughter instead. She is overly happy to teach you, and later on you find out that her father, who has gone missing since she was young, had the same goal of convincing the Ishgardians to grasp onto the Sharlayan way as well.

She takes you to different spots to unlock your connections with the heavens, and gives a history behind the constellations. All in the meanwhile you still have these mercenaries on your tail, and when you end up in Limsa, the Duskwight woman after Leveva even goes as far to try to trick you with glamours thinking it is Leveva you are looking for when she goes missing. Pointing out that glamours do indeed seem to work for race changing!

Eventually Leveva gets granted an audience with the Vault; however this goes not without trouble. Trying to get Forlemont, the Observatorium's boss, to vouch for you is troublesome. Jannequinard is not taken serious by his family at all due to this whoring tendencies, and appears to be disgrace to the name. Forlemont here also calls Leveva a heathen; thinking that the Sharlayan way will distract the Astrologians in Ishgard from their path in their efforts of predicting the attacks of the Dravanian Horde. Oddly enough he does give his seal of approval.

This turns out that he simply tricks you in a trap with the Sharlayans who are looking for Leveva. Jannequinard points out, after you defeat them, that the Tribunal would have him hanged for such, and thus you use this as blackmail to get his 'approval', at least on paper you do.
Talking more with Forlemont you find out the Sharlayans have been plotting this all along, and he aided them due to seeing the benefit of having Jannequinard and Leveva both removed.

Leveva tells you now that more assassination attemps on her have been taken place, even in Ishgard itself. Being unharmed, she explains why she is doing this all; her father, who never returned after he left for his trip, had the same passion of teaching the Ishgardians to way of Sharlayan astrology. Despite never having known him due to her being a newborn when he left, and the objections from her mother, she still felt compelled to follow in his footsteps.

Going out in Coerthas now to spread the good word of Sharlayan astromancy, at first the locals declare it to be heathen. Healing wounded knights leaves the Ishgardian soldiers baffled why they did not embrace the Sharlayan way yet. The knights vouch to get this taught as soon as possible, so they may aid and heal the wounded in battle against the Dravanians.
Here you find out that another Astrologian, a Sharlayan man named Rufin who studied with Jannequinard back in Sharlayan, has been in Falcon's Nest in the past. The man had his throat slit and thrown out to be left for the beasts. People also seem to fear to be branded a heretic and thrown off Witchdrop for supporting the Sharlayan way.

Finally you find out the name of the Duskwight woman who has been following you; Celie. She is indeed send by the Sharlayan ruling body who wants Leveva back for treason; she did not have their permission to do this all. You defeat her and you learn that Mace, the grandfather, saw a vision within the stars and heeded to them to teach the Ishgardians the Sharlayan way, just as his son wanted.
You then find out it was indeed Leveva's father who had himself lynched by a mob of angry peasants.

Continuing your quest to help the Ishgardians, and hopefully knock sense into their minds, you get ambushed by a another Sharlayan astrologian; Sevestre. He tells you he was the one who hired Celie. He is still set onto capturing Leveva, bringing her back, see to her trial and execution; if she indeed taught the Ishgardians "too much". Claiming to have captured Mace, he tries to use him as a way to blackmail Leveva into coming with him, and even going as far to threaten her mother as well. As well you find out Sevestre aspired to be a politician, and was a classmate of Rufin.
He tells Leveva to come to the Answering Quarters if she wishes to see her grandfather again, and in turn be taken back to Sharlayan.
And you save a few more knights, who in turn wish for Sharlayan astromancy to be taught in Ishgard.

In the meanwhile Sevestre stole the dairy of Leveva's father, what points out the connection with Rufin's murder and Sevestre's involvement.

When Leveva decides to show up there, Sevestre backstab you. There is no grandfather, no ship to take Leveva back, instead he seeks to seek to proceed to her execution on the spot.
Knowing powerful magic, Severstre links himself to different Elemental Arcanum in battle. What this does? I have no idea, as I did not fail my AST quest (if anyone did, please let me know!)
Pointed out here as well, that stopping time completely is something what only the most powerful of Astrologians can do.

Defeating him, Leveva makes a pact with him. He lives and he will lie about her whereabouts. He will tell the Forum she has gone missing, and has presumably deceased. Grudgingly he accepts her offer.

Returning to Ishgard you find out that Leveva has cut her hair, and put on another robe, going by the name of Lord Rufin now; acting as if she were her father. Everyone in the Athenaeum thinks no one will fall for it, but they accept it regardless.