In case some of you are unaware, every Cloister thus far has had something to do with a single Destruction Sphere. This is no coincidence, especially if you plan on getting a certain Aeon from the Baaj Temple after the game gives you the airship. At the moment, Zanarkand will not yield a Destruction Sphere - it will LATER, just not now, and you will need it for the Aeon. In any case, if you are coming looking for how to get that Destruction Sphere, you must come back after grabbing the airship; the details for it are just above the section for Battling for the Fayth.
When you enter the Cloister, you’ll notice a diagram appearing on the screen in front of you. This diagram is made up of a number of "Tetris blocks", as I tend to call them. (Play Tetris and you’ll understand.) Also note how there are a number of white dots on the tiles on the floor; stepping on them will make shapes appear. Use this tiled layout in conjunction with the picture below to make the images.
After this, six pedestals will pop out in this, with the next room to the northwest opened up. By pushing in the four pedestals in the northeast, northwest, southwest, and southeast corners, you’ll get more of these "Tetris block" puzzles to solve, just in a larger room. Go there and examine the screen to get the actual puzzle. Again, a diagram of the floor is below - remember to be careful to avoid stepping on unneeded tiles!
After doing this, get the Kilika Sphere from the northern part of the larger room. Take it to the small room and place it in the left pedestal, then go back and take the Besaid Sphere and place it in the other pedestal. The middle of the larger room will begin to shine brightly and a Save Sphere will appear nearby. Be sure to use it. Go to Battling for the Fayth for what’s in the second room.
When you come back to this Cloister after grabbing the airship, you’ll find the panels active again, and they are in the same positions as in the pictures above. In the first room, make the white squares appear (gray in the drawings above for obvious reasons). In the next room, do the same with the squares to grab a Destruction Sphere back in the first room. In the larger room, place it on the slot near the large screen where the Besaid Sphere was to reveal a Magistral Rod chest.