After a number of Espers flew out of the Sealed Gate, it was blocked by a cave-in, and you later crash-landed near Vector. You should go to Vector and visit the palace.
Go to the Palace
Go northeast from the crashed Airship and enter Vector. Heal up at the skinny house (if the woman doesn't heal you, then say No when she asks you to pledge allegiance to the Empire, then defeat the bandits).
Put Terra into your party, and equip someone with the Sprint Shoes relic. Go north to enter the Imperial Palace. Follow the person who comes for you, then talk to the Emperor.
Get Bonuses for Talking to Soldiers
You can get some extra bonuses if you talk to 24 troopers in 4 minutes. To avoid losing time, don't get treasure on the way. You will have to fight some of the soldiers before they listen to you.
Here is a path to talk to all of the soldiers. First, go south and leave the throne room. Talk to the two soldiers and two Magitek Armors as you go south. Then go east when you can. Go up one flight of stairs, then go into the door at the landing and talk to the two soldiers there, then go into the bottom-left room and talk to the soldier there. Go back out and continue going north up the stairs, then go through the door at the end.
Follow the path to go outside. Talk to the Magitek armor to the right, then go left and up the stairs. Talk to the two Magitek Armors outside the door, then go through the door and talk to the soldier. You will have to fight him. He is weak to lightning and water. Then exit the room the way that you came in.
Go down the stairs and talk to the Magitek Armor on the left, as well as the soldier north of it. Then go through the door above the Magitek Armor that you just talked to.
Talk to the soldier at the top of this stair case, then in the next area, go down one flight of stairs and enter the door in the first landing. There are six soldiers here, and one of them fights you. Then go through the northwest door of this room and talk to the soldier. Fight him, then leave the room the way you came in and go back to the flight of stairs.
Go south down the stairs, then go east and south to exit this area. Go down the stairs on the right, and talk to the Magitek Armors. You will have to fight them. They are also weak to Lightning and Water. Finally, go to the southwest part of the area and talk to the guy by the stairs.
You should now have talked to 24 people, so go northeast and through the door to enter the palace, then go north into the throne room and wait behind Gestahl's throne.
During the banquet, you will have to answer some questions. To get bonuses, you will have to answer the questions properly. Here are the answers:
1. Make a toast to our hometowns.
2. Leave Kefka in jail.
3. Doma was inexcusable.
4. Celes is one of us.
5. When he asks if you have any questions, start by asking the top one, then ask another question (the middle), then ask another question (the bottom).
6. Talk about the Espers, and agree that they've gone too far.
7. Remember which question you asked first? Choose that one.
8. Take a break, talk to one of the troopers sitting next to Gestahl, defeat them quickly (they are weak to poison, so use Bio if you have it), and then go back to your seat. (You only have to talk to one of the soldiers to fight them all.)
9. Ask Gestahl to say the war is over.
10. Agree to do Gestahl a favor.
After the banquet, your party will consist of Terra and Locke. Make sure that both of them are equipped.
Go south to leave the throne room. If you got any bonuses, a soldier will walk up to you and tell you what you earned. The bonuses for talking to 24 soldiers and answering correctly during the banquet are as follows: the troops have left South Figaro and Doma, the treasure room in the basement of the Imperial Base to the east has been unlocked, and you will be given a Tintinabar/Tintinnabulum and a Charm Bangle/Ward Bangle.
Now that there is no longer a countdown, you can go around the castle collecting treasure.
When you are done, go southeast from Vector to reach the town of Albrook.