Nobody outside the Temple has anything decent to say or any items for you, so carry on straight to the entrance to the main temple. Before you go inside you'll meet the Luca Goers — your typical washed-up overconfident sporting team. Don't worry, you'll be able to do them in good when you get to Luca and play your first Blitzball match. But for now, watch the scene and afterward head down into the temple. There's nothing of interest in the outside side rooms anyway.
Once you're inside, save, and head over to where Wakka is praying. You should pray as well — it makes everyone like you more, and will introduce you to a new person! Dona is another summoner, although she's a bit snobbish. There are a lot of people you need to ignore for the mean time, aren't there? Once you regain control, walk up the stairs and enter the Cloister of Trials. Tidus will be shoved off the lift on account of not being a guardian. Anyway, one thing leads to the next and you end up outside the proper door to the Cloister of Trials. Choose to go inside.
Kilika Temple: Cloister of Trials
To begin with, take the Kilika Sphere out of the pedestal to the left and put it in the slot by the door. The door will erupt in flames and be replaced by a burning mesh. Remove the Sphere, and it'll crumble. No more door. Head on through to the next room and put the Kilika Sphere in the top wall. A burnt-in Glyph symbol will appear. Remove the Kilika Sphere from the slot and put it in one of the spare ones. Touch the glyph, and the wall will rise.
Head on through. The fire in the channel below will obviously stop you from going any further. Head to the right wall and remove the second Kilika Sphere from it. Take it back to the previous room and place it in the other spare slot. Take the Glyph Sphere from the pedestal in here and put it where you just got the Kilika Sphere from. That section of wall will rise, not only revealing the Destruction Sphere but shutting off the flames as well. Ignore the Destruction Sphere for the mean time and go back to the pedestal. Take a Kilika Sphere from one of the wall slots and put it in the pedestal, then push it onto the shining glyph (sneaky, no?). That whole section of floor will sink into the channel, revealing another Kilika Sphere in a wall slot where the floor was.
Go down and take the Kilika Sphere from the slot and put it in the spare wall slot in the previous room. Take the Destruction Sphere from it's place in the side room and put it where you just got the extra Kilika Sphere from. That section of wall will explode, revealing the Kilika Destruction Sphere treasure, a Red Armlet. This is a piece of armour for Kimahri that rather handily halves fire, ice and thunder elements. Once you've got it, take a Kilika Sphere from the previous room and put it in the door at the top. As before, the door will turn into a flaming mesh, so remove the Kilika Sphere and the remains will disintegrate, allowing you past.
You're now in the Antechamber of Kilika Temple. Talk with your party members who will remark on the fact that you're not supposed to be here. Talk around to learn a bit more about what goes on down here. Kimahri won't talk to you, so just try to leave to trigger Yuna to emerge from the Fayth room. You now have Ifrit! As with all summons, this is the default name that we'll go by, but feel free to change the name if you want. You'll automatically reappear outside Kilika Temple, so leave.
You'll be greeted by more people and a flashback. After more scenes, you can leave Kilika Temple and head back through the woods. Save using the Save Sphere in Kilika Woods or the one in Kilika's inn, then go back to the dock. You'll board another ship — the S.S. Winno this time, and set sail for Luca!