Easy AP
This trick requires a Friend Sphere or Teleport Sphere and Rikku on your team. Use a Teleport Sphere or a Friend Sphere to get to where Rikku starts out on the Sphere Grid. Because she starts out with such low HP, the sphere grid has to give her a lot of HP Spheres. Getting other characters there will add at least 1000 HP.
Collect some Doors Of Tomorrow, either from the first weapon in the Omega Ruins or from the Battle Arena by capturing enemies. Then, gain the ability Overdrive -> AP. Return to the Battle Arena and choose the character who has the weapon with that skill. It is best to have three characters with it for easier levels and to save time. Set the character's Overdrive mode to "Comrade" then choose to fight a King Cactaur. Note: You must have captured all the enemies from the Thunder Plains in order to unlock King Cactaur. In the battle the King Cactaur will use either 10,000 Needles or 99,999 Needles (better). Keep using Phoenix Downs or a Life spell to bring back the character he kills. He will eventually run away and the battle will end with your characters sometimes gaining anywhere from 10 to 20 levels depending on how long the battle lasted and how many times King Cactaur used his attacks.
Get ten Door To Tomorrows, either by bribing the Mech Leaders with 74,000 gil (found in Mount Gagazet), or by stealing from Omega Weapon. A third way to is to catch many different kinds of monsters and then going to the Monster Arena. When you complete one area, the keeper will give you 99 Door To Tommorows. After you have them, customize a weapon with the Overdrive -> AP ability. Next, get the Loner Overdrive by having everyone but the person with the Drive -> AP weapon flee the battle. After a few turns, the character will learn the Overdrive mode Loner automatically. Next, equip the character with armor that can either block or absorb an elemental attack. Engage in a battle with an enemy that only uses that magic (for example, Flame Flan, Snow Flan, Gold Elemental, etc.). Have everyone else flee, and keep Triangle held with a rubber band. In a few hours, you will have 99 S. Levels from one battle.
You need to have control of the airship to do this trick and a weapon with Sensor would be helpful. Find the Omega Ruins with the search. Stay close to the Save Sphere where start at, and move around until you are attacked. If you are attacked by a Master Tonberry have Wakka equip his Capture weapon and use the skill Sleep Buster to put it to sleep. It will sleep for only one turn, so bring in Tidus and cast Hastaga. If you only do magic attacks and no physical attacks, it will use its turn to wake up and you just need to put it back to sleep. Keep casting magic (have Lulu use Demi if she has it) and Sleep Buster. Bring in the person with the Sensor weapon and when it gets down enough where you can kill it, use Wakka's Capture weapon to kill and capture it. You will get the 20,000 AP, but now since its caught you and easy to kill, you can go to the Calm Lands. Go to the Monster Arena and you can fight it again for only 3600, and still get 20,000 AP.
If your characters have the double AP (customize with 20 Megalixers) or triple AP (customize with 50 Wings to Discovery) then go to Bikanal Island (the desert) and complete the Cactuar mini-game so the sandstorm will disappear. Go to where the sandstorm was located and you will encounter either Sand Worms or Cactuars. Sometimes there will be three Cactuars in a battle so you can get 36m000 AP just for winning that fight. With your Double or Triple AP, you receive 72,000 or even 108,000 AP. Just have strong attackers with good agility and accuracy or they will run away or cast 10000 Needles, which can be quite deadly.
Unlike other Final Fantasy games, the AP earned during battles are not divided among the people in the party. Try to as many characters as you can involved to level up faster.
Immediately after you get the Aeon Bahamut, you will be in the Via Purifico (the place where you have to control Yuna because your party was split). Go north to a long hallway lit in red. Go into the hallway, fight a larva person and summon Bahamut. From there, use Thundaga or Firaga. You will get 1000 to 1200 AP. After each battle, go back to the save point to heal your Aeons and party. Repeat until you reach your desired level.
Use the following trick to get more levels and fill your total Sphere Grid within three or less hours. Certain items and weapons are required. First, you need to be able to fight One Eye on the monster arena (in the Calm Lands). Fight and kill One Eye until you get a triple AP weapon with two empty spaces for one of your characters. After you get the weapon, use the Customize Weapon option and add the abilities: Overdrive -> AP (which uses ten Door To Tomorrows), and also either Triple Overdrive (which uses Winning Formula) or Double Overdrive (which uses Underdog's Secret). Once done, change the character's Overdrive to Loner and get into a battle anywhere (preferably Besaid Island). When in the battle, make all other characters leave, and have your character choose the special weapon. After you equipped the weapon, have them skip their turns for at least twenty five turns. Try using Haste on your character and Slowga on the enemies. Then, kill all the enemies. The post-battle screen should state that your character either gained 200,000 AP (if Double Overdrive was used), or 300,000 AP (if Triple Overdrive was used). Within a few hours of gaining these levels quickly, you should be able to complete on the characters entire Sphere Grid.
Complete the Village Of The Cactuars side quest, then equip as many people as possible with weapons that have the Double AP, Triple AP option. Have everyone in your party use a skill. For example, Tidus with Cheer, Yuna with Pray, etc. Have Auron perform Zombie Break on the Cactuars. If it is successful, it does a 100% Overkill.
The best place to train, is at the Sea Of Sorrow. More AP can be gained here than anywhere else, and the enemies are not as difficult as in Omega Ruins.
Have a weapon that has Triple Overdrive, Triple AP, and Overdrive > AP. Set Overdrive to Stoic and fight Don Tonberry. Attack him with the weapon, then he will kill you. Revive that character then repeat the process until you kill him. You will get 30 to 80 Sphere levels.
To get easy AP early in the game, have Tidus learn Cheer. When you battle Sin for the first time on the boat to Kilika, instead of attacking Sin have Tidus use Cheer on himself, Wakka, and Kimahri. Then, repeatedly attack the blue that Sin sends at you. Make sure you use Lulu and Yuna at least once so they also get some AP. If you do this long enough, you can get at least 20 levels up per hour.
