Insanely Fast/Unlimited AP Trick
If you're looking to max out your character's Sphere Grid in a matter of hours there's really no better way for you to do it than through this method. Following this guide you'll get 20+ Sphere Grid levels per battle and you'll spend more time farming Strength, Mana, Speed etc spheres to unlock spots on the grid than you will farming AP.
To get started you'll first need to make weapons with Double AP and Overdrive to AP. You can also farm weapons with Triple AP if you'd like to go that route from One Eye in the arena. This is actually the preferred route for most people and it's the one I would recommend to you.
If you haven't done anything with the Calm Lands Arena yet you'll need to get pretty involved in order to do this AP farming trick. First off you'll need to unlock One Eye so you can farm some Triple AP weapons off of him. Secondly you'll need to unlock Don Tonberry because that's the enemy we're going to be fighting for this AP trick.
Let's start with unlocking One Eye. In order to unlock him you'll need to capture at least 4 of each 'One eye' type fiends. These are the round flying enemies with two stubby legs and a giant eye. Click on the link in this paragraph for a picture of what they look like if you're having trouble figuring out what to kill.
Below are a list of the different flying one eyed creatures that you're expected to capture and where you can find them.
Floating Eye (Mi'ihen Highroad)
Buer (Thunder Plains)
Evil Eye (Macalania Snowfield)
Ahriman (Mount Gagazet)
Floating Death (Omega Ruins)
I highly recommend you just patrol around in each of these selected areas using 3 powerful melee fighters that have "Capture" weapons. If you're unfamiliar with the Monster Arena and how it works I strongly recommend you check out my Monster Arena guide.
Once you've captured all of the Evil Eye enemies you need you should also take a moment to capture all of the enemies you'll need for Don Tonberry. To unlock this guy you'll need to capture one of each type of enemy in the Chamber of Stolen Fayth. The same place you get Yojimbo from. For a list of all the enemies you need to capture inside the Chamber of Stolen Fayth check out my Monster Arena Guide.
The spawn rate in the Chamber of Stolen Fayth is crap for some of the monsters you need and you'll constantly get Magic Urns while fighting here which slow the whole process down a little. If at any point you need to check how many enemies you've left to kill just speak with the guy at the Monster Arena in Calm Lands. When you go through the menu he has available you'll see a number next to the enemies name, this is how many you've captured.
It's a lot of work to unlock Don Tonberry and One Eye but it's totally worth it! Once you've got these two guys unlocked you'll want to start off by fighting One Eye a bunch. He can be a pretty difficult fight if you don't have any Celestial Weapons or a strategy for killing him since he confuses your characters fairly frequently and can be tough to hit with melee for anyone but Wakka.
If you're struggling with One Eye I recommend using Bahamut or one of Yuna's other Aeons to dispatch him. He won't drop a weapon for you every time you win but he does drop them fairly regularly. On top of the Triple AP weapons he'll also drop Magic Defense Spheres which fill empty Sphere Grid slots. This gives you more than one reason to farm One Eye!
After you've gotten enough Triple AP weapons from One Eye it's time to customize them via Weapon Customization and put Overdrive -> AP on them! Alternatively if you haven't gotten someone's Triple AP weapon from defeating One Eye when you go to add Overdrive to AP you'll also want to put Double AP on it.
You could skip fighting One Eye altogether and just put Double AP on everyone's weapon but you miss out on Magic Defense Spheres and it'll also be a hefty Gil sink to get all the Megalixirs you need for Double AP. The easiest way to get Megalixirs (if you need them) is by bribing Varuna. More information about that can be found on the FFX Wikia.
Overdrive to AP is required for this trick!
In order to put Overdrive to AP on a weapon you'll need an item called Door to Tomorrow. For each weapon you want to put this on you'll need 10 of these items. You can bribe Mech Leaders or Mech Scouters for them (more info on that here) or you can unlock Catastrophe in the Monster Arena and the prize for doing that will be 99 Doors to Tomorrow.
In order to unlock Catastrophe you'll need to unlock 6 monsters in the 'Area Conquest' section of the Monster Arena. To do this you'll need to capture 1 of each monster type in 6 different areas. It'll be extremely time consuming and boring for most people but the rewards you get for unlocking new monsters in the Monster Arena are insanely good. The old guy here will fill you up with just about every item you need for maximizing your armor and weapons potential through customization.
Regardless of your decision it's time to add Overdrive to AP to everyone's weapons. Now there is one last thing to do that will be different for everyone. What you need to do is pick two characters that are going to "take all the hits" and switch their Overdrive Mode to Stoic. You'll want to switch everyone else's Overdrive to Comrade.
Stoic means that your Overdrive Meter will charge when you're hit. It's the default setting for Overdrive Meters.
Comrade means that your Overdrive Meter will charge when others are hit.
Don Tonberry uses Karma much like all of the other Tonberry enemies in the game. Karma does damage to your character depending on how many fiends you've landed a killing blow on. This means you'll wanna use one of your primary characters to 'provoke' Don Tonberry. By provoke I mean just whack him with a melee hit, let him use Karma then rez that character and guard on everyone else until his turn comes up again.
Keep doing this trick until Don Tonberry stops walking towards your party. At this point you'll want to flee the battle and collect your AP rewards.
That's all there is to getting a crap ton of AP in Final Fantasy X! The amount I got in that picture may be lower than what you'd get for everyone too - Wakka's overdrive wasn't set properly and Yuna only has a Double AP weapon.