After the talk with Gaepora you can claim another 5 ruppees from the wardrobe. Head outside to finally see Fi. Follow her to the statue of the Goddess. You'll have to do some acrobatics to do that, but eventually you'll reach a hidden entrance behind the statue.
Fi will finally introduce herself (itself?) to you, after which you get to finally pull out the Goddess Sword. The next few minutes are heavy with talk and explanation. At the end of it all you'll receive the Emerald Tablet.
As Fi instructed, stand in front of the crest, charge your sword and shoot off a beam to activate it. Examine the altar to place the tablet and open the path to the first surface area. After all is said and done, there are some preparations to be made before heading out.
First off, as you leave Gaepora's office, you'll receive the Adventure Pouch, a critical item in the game.
First point of order is going to the sparring hall and talking to the Knight Captain. He'll instruct you to get a shield, and you'll be able to get a few things along the way as well. Head to the Bazaar in the middle of Skyloft. There are loads of shops here, you're interested in the one ran by the skinny guy on speed. Buy a shield for 50 ruppees and head to the other side, the potion shop. You'll receive a bottle and can buy a red potion for 20 ruppees.
I am still unsure however how a shield can fit into a pouch. Oh well...
Now you can go back to the sparring hall and receive shield training. Once you're done you're all set to go. Jump off from Skyloft and call your bird. Look for a green column of light and fly to it. You'll be prompted to dive through the opening once you are on top.