To learn Strago's best lore, you must fight an enemy known as Hidon, which can be found on Ebot's Rock/Eboshi Cave.
If you do not have Strago back in your party, fly the Airship to the Cultists' Tower, which is near the center of the world map, inside of a mountain on the narrow piece of land. When you land, be sure to put Relm in your party. Then have Relm talk to Strago to make him join you.
Reveal Ebot's Rock/Eboshi Cave
Ebot's Rock/Eboshi Cave is not accessible at first. To make it appear, put Strago and Relm in your party and go to Thamasa (in the southeast corner of the world map). Walk north, and there will be a cut scene. Afterward, you can go to the island north of Thamasa and enter the cave to find Ebot's Rock/Eboshi Cave.
181. Opinicus/Aspidochelon (weak to Fire and Pearl/Holy)
182. Displayer/Displayr/Mahadeva (use a Fenix Down/Phoenix Down)
183. Hipocampus/Hipocmps/Moonform (weak to Fire and Pearl/Holy)
184. Eland/Creature (weak to Lightning)
185. Cluck/Medusa Chicken (weak to Ice)
186. Warlock (weak to Lightning and Poison)
Make sure that Strago is in your party. Go north and step on the switch to warp. In the next room, there will be a chest above you that likes coral. Go south and step on the switch to warp. At this point, you will have to wander around the dark maze, stepping on warp stones that take you to random locations, while you look for treasure chests. Each chest contains a random number of coral. You have to collect 22 coral before feeding any to the coral-eating chest again. If you try to feed it coral before you have 22 or more pieces, you will just lose coral and the treasure chest won't let you pass.
After you feed it the right number of coral, talk to it again and it will move. Go north until you find Hidon. Talk to it to start the battle.
The goal of this battle is to make Hidon use the special move Grand Train/Grand Delta. It won't use the move until you defeat the four Hidonites/Erebuses. If you know the Ultima spell (learned from Ragnarok Esper or Paladin Shld/Paladin's Shield) use it to kill all four Hidonites/Erebuses at once. Otherwise, the Hidonite/Erebus on the upper-right is the most dangerous because it can inflict Zombie status. Use Fire or Holy/Pearl spells on it. The Hidonite/Erebus on the bottom-right can be killed quickly using any elemental spell. The bottom-center Erebus can be killed with Doom/Death. The bottom-left Hidonite/Erebus absorbs all elements, so use non-elemental spells or just physical attacks.
After getting rid of the four Hidonites/Erebuses, wait for Hidon to cast Grand Train/Grand Delta so that Strago can learn it. After that, you can kill Hidon instantly with a Fenix Down/Phoenix Down. If somehow you kill Hidon before Strago can learn the Lore from it, you can just leave Ebot's Rock/Eboshi Cave and come back, and Hidon will be back.
After Hidon dies, there is a cut scene.