Battles in the Espers' Gathering Place
The Espers' Gathering Place is laid out similarly to Mt. Kolts, alternating between indoor and outdoor mountain areas with different sets of encounters. Indoors, you'll find only two types of Monsters. The Slurms are extremely weak and easy to kill with Fire (or really anything else), and pose little threat. Adamanchyts are a greater threat, with powerful physical attacks and near-perfect physical defense. They may use Sneeze in response to Fight, which is generally a waste of time anyway. Use Magic to take them out.
Outside, you'll fight three different Monsters. The most common and most dangerous of these is the Mandrake. They can mix in the Petrify-causing StoneTouch special with their powerful basic attacks, but the greater threat is from their counterattacks. They may use Raid, a powerful HP-draining attack, in response to any attack sent their way. They will even use Raid in response to the attack that killed them, though the HP restoration will not prevent them from dying. You can exploit their weakness to Fire to defeat them quickly, but avoid using Aqua Rake because they absorb Water damage.
The other outside Monsters are less of a threat. A pair of Insecares sometimes accompany a Mandrake. Like their palette-swapped brethren, they like to cause Berserk with their special, Wing Whisp. This can indirectly trigger Raid counters from the Mandrake and isn't much help if used on Strago. Insecares are vulnerable to Fire like Mandrakes, so a good shot of Fire 2 is highly effective against the group.
Finally, there is the Abolisher. They are strong physical attackers who use Down Dust to inflict Poison when there is only one left. However, patience is highly rewarded in this fight: if you don't kill either of the two Abolishers in a given Encounter, they will use Pearl Wind on their third attack. This is a highly useful healing Lore for Strago, so don't miss this opportunity to get it easily.
Exploring the Dungeon
The layout of the Espers' Gathering Place is a bit confusing. You'll start off in a cavern with three exits. Grab the Heal Rod in the chest near the middle first. You can equip Strago with this and have him attack fellow party members for free healing in a pinch. The north and southwest paths here are a loop that connect to each other. In order to progress, you'll want to follow the partially obscured eastern path. You'll notice Relm following you here and a few other times in the area, but there's nothing you can do about it at the moment.
The path leads to the outside of the mountain, then back in to the Broken Bridge Cavern. For now, all you can do is proceed east, which brings you to the Statue Room and some insight as to why the Espers came here. You'll also be attacked by your old pal Ultros. This is a rare mid-dungeon boss fight, though it doesn't even get boss music. Like your previous encounters, this is a heavily scripted battle.
Ultros's basic attack routine includes Tentacle, Ink, and Stone in addition to basic attacks. All are powerful, though Tentacle is the most dangerous by far. He'll also periodically use Lode Stone to drop a Character's HP to a quarter if the fight drags on. He'll move towards you as he attacks, and if he moves forward 8 times he will use Magnitude8 (which anyone wearing Gaia Gear will absorb) or Aqua Rake. Fight will push him back slowly (once every 8 times), while Magic has four times the pushback effect. Unfortunately, he'll start using his "tri-elemental attack" after being hit with Magic four times. This actually represents a counterattack where he will respond to any of the three basic Elements with the tier-3 attack Spell of that Element. Once you see this message, you'll want to avoid Fire, Ice, and Thunder, even from weapons like the Flame Sabre or Fire Rod. Given that his only weaknesses are Fire and Thunder, this can make things difficult. Aqua Rake is no help either, as he absorbs Water.
In addition to his various attacks and counters, Ultros will cast Haste and Safe on himself after taking at least 6,640 damage. After a further 5,120 damage, Relm joins the fight. This is essentially the end, as she can drive Ultros away with Sketch. With Ultros gone, you can examine the three statues to learn more about what they represent and shed some light on future plot developments.
Relm stays in the party afterward, so update her equipment and assign her an Esper. Her Sketch Command is only situationally useful, as many Monsters resist their own attacks. For example Sketching Adamanchyts usually results in the Wound Ability Blaster, which they resist. It can defeat their Slurm allies, though. The Sketch result for a Mandrake is usually Raid, which is only useful if Relm needs healing.
Finding the Espers
The western door from the Statue Room leads to a save point in a room with no obvious exits. There are three impressions in the ground here, and if you step on them you will fall through the floor to the Broken Bridge Cavern below. These holes will not open if you skipped past Ultros in the Statue Room. The bottom left hole leads onward to the Espers, but the other two lead to Treasures so investigate them first.
The top hole leads to to a path that proceeds both north and west. You'll find a Chocobo Suit to the north and a Tabby Suit to the west. These are the first in a line of costume-style body Armor that can only be used by Strago and Relm, all of which offer nice stat bonuses. I recommend giving the Chocobo Suit to Strago to help offset his naturally low Speed, but the choice is yours. After collecting both Treasures, return to the Broken Bridge Cavern and hop off the broken bridge to the south, then proceed past the Statue Room and return to the Hole Room.
The far right hole leads to a similar platform, this one with only one exit to the east. This leads to an X-Potion. Once again, you'll need to return to the Broken Bridge Cavern and hop off a broken bridge to continue.
When you're ready, dive into the lower left hole. Follow the path north to another outdoor area. This short path leads to the Esper Room where you'll find the Espers you've been looking for. The party will automatically be transported back to Thamasa to meet with General Leo.
Back in Thamasa
After an inevitable turn of events, you'll find yourself in control of General Leo in Thamasa. Talk to Kefka to begin a battle. It doesn't make much difference what you do here, but it is fun to use General Leo's Shock Command. His Offering-powered basic attack is quite powerful as well. You will win in short order, but it's an empty victory. After a series of dramatic events, Celes rejoins the party and you get the Blackjack back.
As soon as you take off, the entire island that housed the Cave to the Sealed Gate will rise up into the air. This is the Floating Continent, your final destination in the World of Balance. You can head there at any time from the Blackjack's controls, but there are still a few things left to do on the surface world. One fun aside to do immediately is return to Thamasa and speak to Strago's rival Gungho, a man wearing red that is now standing in the north part of town.