Final Fantasy IV Side Quest - Getting Bahamut
You can find King of Summons, Bahamut, on the Moon. Like Odin, you must have completed the Cave of Summon Monsters sidequest first. West of the Crystal Lair is a small circular rock. Land on the rocks around it, and go inside. Head south and open the treasure chest for a Samurai Gauntlet. Equip it on Edge, then go north and east. Follow the path to the next floor. Go north and open the western chest for a Samuari armor. Now go all the way east and open the chest for a Samuari helmet. Go north across the narrow pathway and you'll encounter a Behemoth.
Behemoths are tough opponents. Their HP isn't very high, but they have incredibly power when attacked. Use your usual technique. Kain jumps, Cecil attacks, Edge throws, Rosa heals and Rydia summons. She can use Titan, which will do some good damage. Once their defeated, go up north into the next entrance. Go south, and you'll encounter another Behemoth. You'll encounter a few Behemoths along this path, so make sure you brought some Elixirs. Once you get to the end, head north and approach King Bahamut to fight him.
Bahamut doesn't attack until his countdown meter drains to 0. The, he'll use MegaNuke, a very powerful attack that could demolish your party that moment. You can avoid this powerful attack by casting Reflect on all of the party members. MegaNuke will bounce back at Bahamut with Reflect cast. Do the usual attack patterns, but if you want, you can have Kain attack until the countdown is 3 or 2 (or even 1 but it's risky), then Kain should jump, Bahamut will attack, and Kain will jump down and impale Bahamut for major damage. You should have high attack power if you plan to beat him.