FINAL FANTASY X/X-2 HD Remaster

This is the most time-consuming achievement in the game, and should only really be attempted last. You will have to activate every single node on the grid, including blank ones. In order to do this as time-efficiently as possible, you will need to have the option of buying clear spheres from the monster arena trainer. To unlock this option, you will need to capture at least five of each monster in the game. You will need to capture these anyways if you decide to try to beat Nemesis.

The easiest way to get sphere levels to move yourself around the grid is by using the Don Tonberry method. To unlock Don Tonberry, capture one of each fiend in the Stolen Fayth Cavern.

Don Tonberry will counterattack anyone targeting hitting him with Karma. Karma's damage is 100 times the amount of fiends that character has killed, maxing out at 99,999. So, the idea is to turn this damage into XP.

Every character will need to have a weapon that has the trait overdrive->AP. Customizing a weapon with this trait costs 10x Door to Tomorrows. To get 99x Door to Tomorrows, unlock 6 special area creations in the monster arena, then talk to the arena owner.

Although you don't need to, if you further customize this weapon with triple overdrive and triple AP it will expedite things thrice as fast. It will take much longer if you don't have these traits. Double overdrive and double AP are also beneficial if you can't get the triple traits. Even if you only have a overdrive->AP weapons without triple traits you will get a decent handful of sphere levels.

EDIT: Alternatively: The amount of AP karma gives is based on how much, or rather little, HP the attacker has. If you attack with someone who has little to no HP spheres activated, it will only take two attacks from Karma to fill 99 sphere levels for everyone with only the overdrive->AP trait. It will probably take more than this, however, because realistically at this point in the game even your weakest character will have some HP. What is problematic is that Karma's damage is based on amount of fiends killed by the attacker. A large amount of friends killed will again usually mean that that character will already have a decent amount of HP. To get around this, you can have your weak character farm dingos in the monster arena until they have 999 kills, and avoid filling HP spheres on that character while everyone else gets their grids filled. If you don't want to bother with either having or making a character have little to no HP, then just use triple trait weapons (even though it is slower). Credit to @CLQL8TD MADMAN for this information.

Set the overdrive mode for whoever is attacking Don Tonberry to Stoic, and the rest of the party's to Comrade. Now, skip the turns of everyone but the one who is attacking Don Tonberry (Y button). The character who has killed the most fiends should attack Don Tonberry. They get hit with Karma, and the entire party immediately gets AP. Revive them and repeat until Don Tonberry stops using Karma, and then Flee. If everyone has triple AP and Overdrive weapons, they will all get 99 sphere levels (assuming Karma is at max damage).

It's helpful to equip someone with Auto-Phoenix armor to save some time reviving the attacker, although not necessary.

Now that you have sphere levels to move around, you will need spheres to 1) activate spheres and 2) fill empty nodes. The fastest way to do this will require one character with a break damage limit weapon that can also hit for 90,000 damage. All fully upgraded celestial weapons have break damage limit, or you can customize one of your own.

Kottos drops the most activation spheres in the game along with Fafnir (but Kottos is much weaker). He is unlocked by capturing one of each fiend from Mi'hen Highroad. At the beginning of the battle, use the distiller for whatever sphere you want Kottos to drop. You can buy distillers from the arena owner. When you overkill Kottos (he has 450,000 HP), he will drop forty of those spheres. There is a very low chance that he will only drop 2, but that rarely happens. The easiest way to kill him is to hit him with one Attack Reels overdrive from a max damage Wakka. Again, it's helfpul to have Auto-Phoenix armor to save time reviving characters.

The easiest way to fill empty nodes is to repeatedly kill One-Eye, who is unlocked by capturing one of each eye-type fiend (Floating Eye in Mi'hen Highroad, Buer in Thunder Plains, Evil Eye in Macalania Snowfield, and Ahriman in Mount Gagazet). Overkilling One-Eye will drop 2 MP Defense spheres that you can fill nodes with. There are other monsters in the arena that drop other types of spheres, but One-Eye is the easiest to kill at 150,000 HP. Vidatu drops MP spheres and only has 95,000 health but has a very high evasion and will probably take more turns to hit than One-Eye. One-Eye also drops Triple AP weapons which you can further customize with Overdrive->AP for Don Tonberry. Or, you can sell them to the arena owner for a pretty good amount of gil.

With the above three monsters you will be able to reliably fill out the entire grid... except for luck spheres. The fortune spheres used to activate luck spheres are only dropped from Earth Eaters in the Original Creations section of the Monster Arena. This is where the clear spheres that you buy from the Arena owner come in. Since these battles are very time-consuming, I avoided them entirely and used the clear spheres to erase every luck sphere on the grid. I then replaced them with the MP Defense spheres from One-Eye. I don't recommend doing this if you plan on killing all the superbosses without Zanmato, as you need luck in order to hit many of those enemies. I was still able to beat both Penance and Nemesis with no luck spheres on the grid by using Zanmato.

Please comment with any questions. Thanks for reading!

EDIT: If you haven't started the game already, you could pick the expert sphere grid as opposed to the standard, as there are less spheres to fill and should take less time to grind the XP necessary. However, if you haven't played FFX before, I wouldn't recommend this. Each character on the standard sphere grid starts as a sort of "class" that you can use to exploit certain enemy weaknesses (Wakka for evasive enemies, Auron for armored). On the expert sphere grid, if you don't know the direction each of your characters are going you could end up with a character with no healing spells, or in a fight that would be easy normally, but are disadvantaged in on expert. If you feel like you have a good handle on character mechanics then it's no problem.