Matt Bird writes all sorts of nonsense, but he dedicates a large chunk of his time to writing game walkthroughs.
The Returners are getting close. They’ve trounced the best the Magitek Factory has to offer, and with two new pieces of magicite under their belts they're standing on the doorstep of the Magitek Research Facility. Maybe the next few espers will be in better shape than poor Ifrit and Shiva.
The door next to the restroom in the pit where you fought Ifrit and Shiva leads to a set of stairs. There are some unique enemies here you won't be able to fight elsewhere, so get into three or four battles to make sure you get them all for the Veldt. The next room, which is officially the Magtiek Research Facility, is much the same. Check beside the left-most tube on the bottom row to find a hidden Stoneblade before proceeding.
In the next room, you'll find a pale figure standing in front of the door to the Magitek Research Facility's draining tanks. Talk to it for a boss battle.
If you haven't been gaining new magic, Number 024 can be a real pain. After several rounds of normal attacks, it will change its elemental type, forcing you to adjust your attacks so as not to heal it. The easy way around this is to blast it, right from the start, with the spells you gained from Ifrit, Shiva and Ramuh. Number 024 will take roughly 1,000 HP of damage with each blast, and within two rounds of a constant battering will go down without much of a fight. If Number 024 does change its elemental type, use Libra to discover its new weakness and attack accordingly (or stick to non-elemental special or normal attacks). You can steal a Blood Sword from Number 024 if you're lucky, though you're much more likely to get a Rune Blade. You'll get an Icebrand for beating Number 024.
In the next room you'll find the Empire's extraction tubes, each containing an esper. Hit the switch in the north of the room to release the espers, who, each on the verge of death, turn themselves into magicite. You'll receive six shards: Bismarck, Carbuncle, Maduin, Phantom, Catoblepas and Unicorn. You'll also confront Cid, pioneer of the Magitek extraction process, and Kefka. Celes, accused of betraying the Returners, will spirit Kefka away and buy you the time to escape… and Cid, apparently with a change of heart, will lead the way. One member short, hop onto the lift with Cid.
Save at the bottom and talk to Cid. He'll reveal more about Celes' past and knock you onto a nearby mine cart. The next sequence is a first-person escape through the tunnels beneath the Magitek Research Facility. You'll get into fights against Magna Roaders along the way, though they're easier than the enemies inside the facility. If you're looking for a complete game experience, you'll have to fight the two varieties: the smaller red Magna Roaders and the larger purple Magna Roaders. The red ones are quite rare, so don't be surprised if you have to restart the section a few times to see one.
Eventually, you'll hit a new enemy: Number 128.
Number 128, Left Blade and Right Blade
Number 128 is not as tough as it looks. This beast will use physical attacks from its two Blades, as well as Net to stop a single character. You can destroy the Blades to stop physical attacks, but doing so will push Number 128 to use Gale Cut on every member of the party. Destroy one of the Blades, ignore the other, and focus most of your firepower on the main body. Use powerful special attacks and Fire or Fira to knock down its HP. Avoid normal physical attacks, or Number 128 will counter with Blood Feast, sapping some of your HP. Also avoid ice attacks, as Number 128 will absorb the ice element. Do your best to Steal from it, as you can gain a valuable Kazekiri for Cyan. Not a terribly difficult boss and you get a Tent for winning.
After beating Number 128 you'll wind up outside the Magitek Research Facility. Use the save point north of where you started and head back to the entrance of Vector, fighting off any random battles along the way. (Linger here until you face the Chaser enemy.) Setzer will greet you and usher you back to the airship … though Kefka's got a little surprise waiting. Your second trip on an airship is decidedly more violent than the first.
Despite looking largely identical, the two Cranes are different. The one on the left resists and uses lightning-based spells, while the one on the right resists and uses fire-based spells. Use too many of either on the respective Crane and you'll get hit by a fantastically-painful spell. Avoid elemental attacks altogether and mangle one Crane at a time with your strongest attacks. Setzer joins you in this battle, and his Slots command can come in really handy if you manage to hit it big with an all-hitting attack. The Cranes don’t hit that hard and have little HP, so this fight should be over soon after it begins.
Once the Cranes are down, Setzer will steer the airship back to Zozo to check on Terra. You'll learn much about her history and her true lineage, and she'll rejoin your team. You'll also gain both Setzer and the Blackjack, significant boons in the battle against the Empire. Things are looking up!