To do this quest, you need to have met two prerequisites. First is recruiting Yuffie, which is a separate achievement.
The second is either completed the Wutai sidequest OR you're about to head into the Northern Crater that functions as the final area, in which case you can skip the next section. The Wutai sidequest can be done as soon as you get the Tiny Bronco, and a guide will be listed in the following spoiler section.
Note that if you want all four Enemy Skill materia to have all available spells, you will need to wait until you have the Highwind to get the last one from the Chocobo Sage, as this will be the last place you can learn Trine.
Now for the Pagoda sidequest. This is a series of five 1v1 boss battles as Yuffie ascends the tower. You can leave, rest, and save between each fight. For equipment, at minimum, you should have the Razor Wing as your weapon, which is bought in Wutai. Alternate weapons are:
Magic Shuriken: Free inn at Wutai. Check the wall on the left for a sheet curtain near the bottom corner and follow the passage. NO LINKED SLOTS FOR ELEMENTAL ATTACKS.
Spiral Shuriken: Buy in Costa del Sol after getting the Highwind.
Crystal Cross: Buy in Mideel after getting the Highwind.
Oritsuru: Get the Leviathan Scales from the Submarine Dock, then go to the cave on the Da-Chao path and interact with the fires.
Conformer: Found in Sunken Gelnika
Equip the best armor you have, which ideally has a linked slot for Elemental, and the accessory should either be tailored to the fight or a Ribbon. Materia, you should equip whichever of the following you have: HP Plus, Counter Attack, Restore, Enemy Skill. For any use of Elemental, equip the highest ranked copy to your armor, and a potential second copy to your weapon, where rank doesn't matter. Finally, make sure Yuffie is in the back row, as she should never be in front.
Remember to save and heal between each fight so you can bring your strongest summons, spells, and enemy skills to speed it up.FIGHT 1: GORKI
Gorki is only challenging if you're fighting him before the Highwind. Equip Gravity-Elemental to your armor and ChocoMog-Elemental to your weapon if possible, otherwise just ChocoMog as he is weak to Wind. Destruct is optional if it's rank 2 and you have DeSpell to remove his barriers.FIGHT 2: SHAKE
After Gorki, this is a generic enemy with 4,000 HP. No particular strategy, just beat them senseless.FIGHT 3: CHEKHOV
If you don't have a Ribbon, equip the Jem Ring that was dropped from the Materia Keeper fight. This prevents Chekhov from paralyzing you, and results in another generic 5,000 HP enemy.FIGHT 4: STANIV
I'm running out of ways to say "this is a generic enemy with 6,000 HP."FIGHT 5: GODO
If you don't have a Ribbon, you'll need to equip a Headband, which can be bought in Gongaga, and stock up on Cornucopia (Cosmo Canyon), Remedy (Mideel), Esuna through Heal materia, or Mini through Transform materia. Set up Lightning-Elemental armor, and either a Gravity materia with at least Demi2, or an Enemy Skill with Laser. If you have any linked slots left in your armor, add Poison-Elemental or Gravity-Elemental if possible.
As mentioned above, if you are looking to max out all Enemy Skill materia, this is the only place you can get Trine on your fourth Enemy Skill, so if that's important to you, wait until you have them all before you continue.
Godo has 10,000 HP. Open the fight with your strongest move, either Demi3 (7,500 damage), Demi2, or Laser (5,000). From here, hit with everything hard you can. Once you drop him to 4,000 HP, he will counter every hit with Cure2, so you must hit for at least 1,300 damage every time you attack. Poisoning does not trigger the heal reaction if you need to wait it out.
He can cast Mini and Sleepel, which are countered by your equipment/item stash/magic options. He can also use Demi3 for 75% of your current HP, Bio2 which has a good chance of poisoning you, and Trine as a lightning attack as the most deadly.
Once the fight is over, you'll receive the Leviathan materia.