What exactly do you want to know? It’s long but not very complicated. It’s mostly 3 parts.
Preparation phase. You get 1-2 quests here that might come from a larger pool but from what i’ve observed the most common are increasing minor population count (cooperative in the entire galaxy) and having the best trade route (in dust production) 20 turns later. The rewards are not terribly exciting considering how late in the game those quests start and heroes are not yet locked so you can sort of ignore those.
The first part of the academy quest itself start after you choose your side and your team have to discover more lodestone systems than the opposing team. To “discover” a lodestone you just have to move a hero led fleet there. Seeker admirals work wonder for that quest as those systems are off-lane. You won’t be able to see those systems before the quest start so you can’t prepare besides having several admirals in fast moving fleets. It’s pretty quick and the lodestone link improvement unlocked is quite nice giving you bonus to all FIDSI (more on smaller planets)
The last part is where you will actually fight for control of the lodestones. No matter who won the fist part, everyone starts equal for that final part. You’ll have to orbit lodestone systems with a hero-led fleet for 10 turns to gain control of that lodestone for your team and having even a single enemy scout in the system will stop the progress but you can attack you friends and even allies in lodestone systems if they are in the opposing team. This doesn’t have any diplomatic impact which is both practical and and lame. I think that quest should have more impact on the diplomatic state of the galaxy as currently it’s “oh, you’re trying to bring back dead gods while i’m attempting to maintain the galaxy in a state i know allows our races to survive so i’ve destroyed your whole fleet … now to the important stuff, let’s discuss the tax on Jadonyx exportation”
As soon as a team have taken control of more than 1/2 the lodestones, that team wins and the rewards are awarded based on how well you contributed. Playing on impossible i usually have no issues being best, sometimes only, contributor for my team.
Additional Note: All members of the losing team, no matter restoration(a) or preservers(b), will lose all access to new hero’s whether market or academy. Only way around that might be some of the special quests that can result in getting special hero’s, like ‘Petal’ the pirate lord, but, I’m usure of this as I’ve normally either done the quest before the academy war, or won a final victory before they show.