Cloud Strife:
You gain the Omnislash manual for Cloud in the Gold Saucer Battle Square. You'll need to take plenty of gold with you, at least 200 to be safe, and fight until you earn 32,000 battle points. You can then buy the manual as an item.

Barret Wallace:
You can get the Catastrophe manual for Barret any time after stopping the train in Corel on disk 2. Once this part of the game is done, enter the house closest to the inn on the upper tier. There is a lady inside who says something about her late husband and will give you a memento of the man, which is the Catastrophe manual.

Tifa Lockheart:
Tifa's Final Heaven manual can be gained on disk 2 or later, anytime after getting Cloud back from his coma. You simply return to Nibelheim with Tifa in your party, and make your way to her room in her house. You want to play the piano there. You'll need to remember the tune you played in the flashback. If you can't recall, it goes like this: X, Square, Triangle, R1 + Triangle, R1 + Square, X, Square, Triangle, R1 + X, O, X, Square, X. If you did it correctly she should say something about having the sheet music for the song and you'll also get her Final Heaven manual.

Aeris Gainsborough:
When you get to Junon you now want to head north east to the river. There is a shallow spot you can cross using the buggy. Do so and head east until you find a cave entrance. Inside the cave you will find a sleeping old man. He won't talk to you much until you battle a certain number of times. The trick is to make the last two digits of the number of battles you fought match. For example, when you've fought 33 battles or 244 battle he will talk to you since the last 2 digits match. He gives you a bolt ring or mythril. If you want Aeris limit break you need the mythril. From the Gold Saucer near the end of the river you'll see a little house on your right. The guy inside will talk about the keystone for a bit. Give him the mythril and he'll reward you by letting you pick a chest as a prize. Go up the stairs to the top floor and go all the way left. Open the little box on the wall to receive the Great Gospel manual. The big box on the lower floor just contains a Gold Armlet.

Red XIII:
You get Red XIII's Cosmos Memory pretty early in the game, on disk 1. The first time you get to Nibelheim you'll be heading to the Shinra mansion. You'll need to open the safe on the 2nd floor to get the Odin materia and the basement key that you'll need to find Vincent. When you open the safe you'll fight a boss called the Lost Number. For winning this fight you'll receive the Cosmos Memory manual.

Yuffie Kisaragi:
You can get Yuffie's All Creation Manual as early as disk 1 when you have the Tiny Bronco. Simply travel to Wutai and challenge the masters of the Pagoda. Once you beat the last boss, who is Godo, Yuffie's father, you'll receive the Leviathan summon and the All Creation Manual.

Cait Sith:
He has no level 4 limit break and therefore no manual to teach it to him.

Vincent Valentine:
You can get Vincent's Chaos manual as soon as you gain access to the submarine. Simply put him in your party and then take the sub to the river mouth north of Cost del Sol. Make sure not to miss it altogether. Dive underwater and look for a small cave entrance. Just follow the path until you come to a circular room. Surface and you'll be in a small lake. You can dock the sub on the side and go into a cave hidden behind the waterfall. You'll see a flashback sequence with Vincent here. Once its over leave the area, and go somewhere that you can fight. I suggest the Gold Saucer area since it's the closest sub dock. After you've fought at least 10 times return to the cave where he had his flashback and you'll receive the Chaos manual and his final weapon, the Death Penalty. When looking for the undewater tunnel be aware that it branches off of the tunnel that leads you to the Key of Ancients. You can also use a green, black or gold chocobo to do this in place of the sub.

Cid Highwind:
Cid's Highwind manual is easy to get. Once you have the sub travel to the Gold Saucer sub dock. Now you want to dive underwater. There will be the wreckage of a cargo plane nearby. You can find all kinds of goodies in this plane, called the Gelnika. One of the chests in the cargo room contains the Highwind manual.