When you arrive and regain control, examine the nearby terminal to ride the stairs to the bottom. After a brief scene, head into the next area and get the Al Bhed Primer Vol. XXI I from the ground, then save and continue into the Cloister proper.
In Bevelle, you’ll basically move around on glyphs atop some kind of electromagnetic current. They move automatically in a set direction, but you can choose where to go at the junctions. To do so, you will need to press X while the glyph at the intersection glows, pointing in the desired direction. So long as you make a map or something, you ought to be fine. Or you can follow my Guide. Anyhow, I tell you this now because it doesn’t easily stop, and it’s hard to look back and forth between a guide and the game.
Begin by examining the nearby pedestal to make a glyph platform and ride it to the intersection. Go right there, then grab the Bevelle Sphere at the end of the path. (It will not be on the pedestal; it’s the one you just reached.) Grab it and put it, too, on the pedestal, then push the pedestal back into the current. Ride it and you’ll make a right at the next junction automatically, but need to take another right manually. Once you do, put a Bevelle Sphere into the nearby recess.
Push the pedestal onto the current again and ride it for a loop back to the start. Go straight for the first intersection, and left at the second, then take the Bevelle Sphere from the wall and put it on the pedestal. Return into the current and loop back to the start. Go right at the first fork and left at the next. Go straight at the next two and, at the final junction of this "floor", go right. Put a Bevelle Sphere into the recess on the wall, then return the pedestal into the current. Loop back to the start.
Go right at the second intersection next (discounting the one you begin with) and you’ll find a Glyph Sphere. Grab it, put it on the pedestal, and return into the current. Continue to the next junction and take a right (even though your GPS likely says left Hehe). Run across the solid path to the glyph, where you should place your newly-found Glyph Sphere, to find a Destruction Sphere . Take it and put it on the pedestal.
Yet again, push the pedestal into the current and loop to the start, then take a right at the second intersection so you can place the Destruction Sphere where the Glyph Sphere started at. Go into the current and take the next right, then snatch up the Bevelle Sphere you put here earlier. Place it on the pedestal and use the current to make another loop. Go right at the first intersection (discounting the starting one). Push the pedestal across when you land so you can move further.
After landing, take a Bevelle Sphere from the pedestal and go upstairs. At the top, open the chest for an HP Sphere . Step on the sparkling floor panel nearby to get your pedestal as well. Push the pedestal into the current to the left to get the chest with the Knight Lance .