The Cloister of Trials in Kilika is centered around fire and if you’ve played previous Final Fantasy titles, then you probably have a good guess who the Aeon might be here. Upon entering the first room with something in it, look for the pedestal on the left side and take the Kilika Sphere out of it. Looking at the door in front of you, there will be a sphere-shaped recess on the right side of it, which is where you will put the Kilika Sphere. Doing this will ignite the door, but the fire will persist, so take the sphere back out of the recess to be able to continue.
In the next room, head straight to the wall directly across from the entrance and place the Kilika Sphere in the recess there. This will cause a glyph to appear on the northern wall, but take the Kilika Sphere out of the same recess and place it on either the left or right wall. Touch the glyph you spawned before and this will open up the way to the third room, where you will find a wall of fire blocking your way. On the right side of this room will be another Kilika Sphere, so take it out of the wall. Return to the previous room and place that second Kilika Sphere on the wall opposite the first one.
In the second room, there is a pedestal with a Glyph Sphere in it, so grab it and place it in the recess in the third room, where you grabbed the second Kilika Sphere. This will not only open the wall behind the recess, but it will also remove the wall of fire. Double back to the second room and take either of the two Kilika Spheres, placing it in the pedestal where the Glyph Sphere was located. With it in there, push the pedestal into the third room and onto the glowing spot on the right side. Doing this will lower the piece of floor the pedestal is on, revealing another Kilika Sphere.
Return that third Kilika Sphere to the second room, in whichever recess you took the second out of before. That’s all of that out of the way, which is to help you nab the treasure in this Cloister of Trials. In the third room, in the alcove that opened up earlier, there will be the Destruction Sphere. Take it and place it in the recess where the third Kilika Sphere was, blasting open the wall right there and allowing you to grab the temple treasure, a Red Armlet. Grab either of the Kilika Spheres from the second room and place it in the recess at the far end of the third room, then take it out. Enter the inner chamber for a scene, then speak to each of the party members there. As you attempt to leave, a cutscene will occur and you will be able to name your second Aeon (default name is Ifrit). When you regain control, exit the temple.