Note: Possibly THE hardest Monster to encounter in the game, the Goblin Prince can be located anywhere on the entrance floor of Kai'son Castle. He has a very low chance of appearing, but just walk around and get into Battles until you find him.
As you enter the castle, head north past the Sunfire, since you can't get it yet and go up to the gate and use the Goddess's Bell on it to unlock it. Then, enter the door and in this area, follow the path to find Gordon and talk to him, as he explains how he thought only he could get the Sunfire, though the Monsters were too tough for him. He needs the EGil's Torch to hold the Sunfire and Maria asks if he knows his way around the place, but he truly doesn't. He asks to join you, as you allow it, though, don't bother training Gordon, though, I do recommend getting his HP up a little, since he's very weak as it stands. Give him all of Josef's Equipment and if you still have it, the Mythril Spear you picked up earlier.
Once you're ready, head east and follow the path down the stairs and back in this area, head south to find a Treasure Chest, which holds a Cure Scroll. Then, head back north and go west now, then head south when you can, ignoring the door, since it leads to an empty room and head up the stairs you come across. On this floor, head south and then east when you can and follow the path and grab the Treasure Chest you come across to get 300 Gil. Next, head west and follow the path up the stairs and on this floor, follow the path and head east when you can, then head south down the middle path in the room and head west to find a Treasure Chest. Open it to get a Gold Shield, but you'll have to fight some Wererats for it.
After you're done, I suggest giving the Gold Shield to Firion if you have him with a Shield or anyone else you wish, since it guards against Poison and has the same Defence as a Mythril Shield. Next, backtrack to the north and head east, then south and follow the path and go up the stairs into the next area. On this floor, head west past the first path to the south and go down the second path to the south and enter the door near you. In this room, grab the 3 Treasure Chests here, which hold a Mythril Sword, Mythril Axe and a Werebane Staff. Then, exit the room and head southeast and follow the path until you find some stairs. Go up them and on this floor, head northeast and head south when you can to find a Treasure Chest, which holds a Golden Mail, plus a Battle with some Mines. I suggest you give the Golden Mail to Firion or whoever you want with the best Defence at the moment.
Next, head northeast and enter the door you see and in this room, open the 3 Treasure Chests here to get an Eye Drops, Mallet and an Antidote. After this, exit the room and head as far southwest as you can to find some stairs. Head down them and back on this floor, enter the door and prepare to fight a Boss, so, heal up, get ready and approach Shrieker and talk to him. If you want a strategy for this Boss, head to Section VII. Boss Guide now, otherwise, fight him yourself and then continue on. After the Battle, grab the Treasure Chest nearby to get the EGil's Torch, then either walk back out to the front or use Warp if you have it, then re-enter the castle and go up to the Sunfire. Use the EGil's Torch on it after you examine it and you'll obtain the Sunfire. Next, leave Kai'son Castle and outside now, head south to see a scene where the Dreadnought is chasing Cid's Airship.
After the cool cutscene of the Airship being captured, you'll need to make your way back to Altair now. It's a long ways back, but to make this easier on you, head south and look for a small circular clearing in the forest and enter it. In this area, talk to the lone Chocobo and then get on it and ride it back to Bafsk, which is to the northwest from where you are. From Bafsk, head to Poft and once there, ride the Ship to Paloom, then make your way back to Altair. Once in Altair, sell off any Items you don't need, buy anything you may need, heal at the Inn, then go and see Hilda… who's missing. Huh, weird. Ok, instead, go and see the King and Mindu. Talk to Mindu to learn about what happened to Hilda, then ask him about the Dreadnought. He tells you that it needs to land and refuel sometime, but that's all. Hmm… not much else to go on. Go to the main room and talk to the people here to learn about that the Dreadnought may be up north past Fynn.
Well, it's all you have to go off on, so, when you're ready, leave Altair and outside, if you wish, train a little more. weapons should be anywhere from Level 4-6, same with your Shield, Magic should be about 4-6 hopefully as well and HP should be 500-600 or so. Stats can vary, but whatever they are, depending on if the rest of the above is around there, you'll be fine. Just make sure you have the best Mythril Equipment possible, plus the few pieces of Equipment that are better than Mythril that you recently received at Kai'son Castle. When you're ready, head up to Fynn and head northwest, until you finally reach the Dreadnought after walking for a good while. (Note: New Monsters are in the area, mainly the ones I listed back at the start of the game near Gatrea and Fynn that I told you to fight so that you could get an INSANE amount of Gil and Mythril Equipment early on.) When you're ready, though, enter the Dreadnought.