Cloud Strife: Omnislash To get Cloud's ultimate limit break, go to the Gold Saucer and then go to the battle arena. Then you must go to the Battle Arena and get 32000 battle points. To do this, you must not leave the Battle Arena, and I recommend having at least 100 GP.
Barret: Catastrophe You must stop the train carrying the huge materia from crashing into North Corel. Go to the tent at the Northern edge. A woman with a hat on will give you the Catastrophe item.
Tifa: Final Heaven Well, you need to play Tifa's Song on the piano at her house. Not only will you get her limit, you'll also get a note from Zangan, her trainer. The notes you have to play is X, square, triangle, R1 + triangle, R1 + Square, X, square, triangle, R1 + X, Circle, X, Square, X
Red XIII: Cosmo Memory Go to the Old Shinra Mansion in Nibelheim. Go upstairs and to the safe. Enter the code Right 39, Left 10, Right 59, and Right 97. You will get the Cosmo Memory item after you defeat the boss. You will also get the basement key, which lets you get a secret character. Vincent!
Cait Sith: Slots To get this, you must simply kill 80 enemies.
Cid: Highwind To get this, go to the downed Shinra plane under water. Stay toward the west and you should find the plane. Inside, go to the cargo hold and the Highwind is lying near some boxes.
Vincent: Go to Lucrecia's cave in disk 3. There you should find his limit break and Ultimate Weapon.
Aeris: Great Gospel To get this, you must go to the sleeping man's cave in the 1st disk. He'll give you a mythril. Then go to a man sorta near Corel or Costa Del Sol. He will let you pick out of the small or big box. Pick out of the small metal box upstairs and you will get Aeris's Great Gospel item.
Yuffie:All Creation You go to Wutai the island group to the south west or bottom left of the map. You go to the back of this town and visit the big tower or Godo's Pagoda. Defeat all the 5 competetors and you will get the All Creation item and as a bonus, you'll get the Leviathan Summon Materia.