The Cloister of Trials in regards to FFX is simply that: a set of trials that is meant to keep the unworthy from learning the all-powerful abilities to summon Aeons. (Though you won’t get anything truly worthwhile until the last 1/4 of the game.) As far as you ought to care, it’s a puzzle. They get more annoying in later areas.
Anyhow, go north and examine the glyph on the wall before you; upon touching it, a path opens up nearby, so follow it. Examine the wall at the bottom and you can learn about the different spheres: the green Glyph Spheres that are used mostly to open doors and such, the purple Destruction Spheres that remove obstacles (often optional ones for treasure!), and then the " Place " Spheres (i.e. Besaid Spheres) that have effects depending on where you are.
Grab the Glyph Sphere after the brief tutorial, then go downstairs and shove it into the small space on the door at the bottom. Go through the newly-opened door, then re-examine the Sphere to take it back. (FYI, you can only hold one at a given time. Odd, as you have two hands and probably some kind of pockets, but whatever.) Go along the hallway and place the Glyph Sphere on the pedestal near the left wall. This will reveal a Destruction Sphere but leave it for now. Continue along to the glyph on the wall and touch it, then take the Besaid Sphere you find.
Pick up that Sphere and put it on the pedestal just to the left. Return to the Destruction Sphere you found moments ago and place it where the Besaid Sphere was to open a secret room; beyond is a Rod of Wisdom , a pretty decent weapon for Yuna, who will join our party shortly. (It boosts Magic by +8% and has Sensor.) Return to the pedestal again, this time pushing it into the center of the room (just run into it with the Analog Stick): it will sink into the floor and make you an elevator. There will be a lengthy scene thereafter so enjoy.
IMPORTANT : If you wish to acquire Anima later in the story, you must collect the treasure in each Cloister by using the Destruction Sphere. Do not skip that part if you want Anima!