The game starts off with Link's nightmare. After you get awakened, take the money out of the wardrobe next to the door and head out. In the academy you can talk to the various characters to learn control basics. Once you're content with that, head out through the main door into Skyloft.
Head left for yet another tutorial. Sprint up the crates near the wall and jump over. After reading the instructions for rescuing the cat, sprint up and grab the ledge at the wall in front of you. You can shake the Wiimote in one direction to leap, but it costs stamina, so be careful. Sprint jump acros the next gap and make your way up the vines. Again you can flick the Wii remote to climb faster. There's a box a few meters ahead, push it against the wall and climb up. Grab the cat from the roof and safely take it back down. Unfortunately, you cannot toss it down the roof.
The game will show you the direction to the Goddess statue, head that way. On your way you can learn about saving the game and pick up a stamina fruit.
As you climb the stairs you'll find another stamina fruit. Head forward to start a cutscene with Zelda. During the cutscene you'll be prompted for a conversation choice, although it doesn't make any difference in the long run. Once the conversation is done, Zelda will push Link off the edge to "help him practice" (Yeah, right). Moments later, she'll be saving his ass from a gruesome death.
Your next task is to find your Loftwing, but before that you'll have to go to Instructor Horwell and let him know about your missing bird. Go all the way back to the academy, but make sure to talk to Horwell on your way back (The guy who sent you to fetch the cat). Once back inside, go talk to Pipit (Lad in yellow tunic). He'll instruct you to go to the plaza. Head out the door you went out through at the very begining and continue forward through the gate.
You'll get a quick tutorial about rolling, continue onward to the plaza (Wide open area). Approach the trio near the grass patch to trigger a cutscene.
Once Groose and his lackeys leave and you get back control of your character, go back to the Knight Academy, where Pipit was. You'll find a girl in his stead, telling you that he was looking for you and is now in front of the sparring hall. Just exit the academy and go north to reach said sparring hall. After the conversation, enter the sparring hall and talk to the Knight Commander. He'll instruct you to get the Practice Sword from the back room.
You now have the option to do a sword tutorial. Once you're done, head out of the sparring hall to trigger another conversation. Once that is done, you can finally leave.
Now you need to reach the north-east cave, marked on your map. You can either get there going through the bazaar, and making your way around Skyloft, or you can go to the second floor of the academy and hang a right at the gate near the stairs that lead to the Goddess Statue. Jump across the stones to reach the other side and go up the path leading to the cave. Slice open the entrance and get ready for your first enemy encounters.
The journey through the cave is pretty straightforward. Once you clear the wall you have to dash up on, look for vines on your left. Climb up to reach a chest with 20 ruppees in it. In the last room before the exit, look for a hole hidden in the grass patch on the right. Inside of it you'll find another 40 ruppees. Once outside you can refill any lost hearts using the flowers to your left. Head forward to trigger a cutscene with Zelda arriving. Apparently she's on drugs because she hears voices in her head.
After regaining control, continue onward and you'll find your Loftwing. In order to free it, you have to cut all the ropes holding the makeshift cage together. As you might have guessed, you need to use precise swings using your Wii Motion Plus to do that.
Once the bird is free, and all the talk is done, you'll get basic instructions on flying. Right after all the talk you'll be thrust into the Wing Ceremony. Dash off the ledge and press down on the D-pad to call your bird. You now have to catch the yellow Loftwing carrying the statue. You'll get a prompt to snatch it once you're close enough, but due to Groose's "fair race" you'll have to do it twice.
The eventual win will get you to the ceremony with Zelda, at which you'll get the sailcloth, which smells nice.
Right afterwards, Zelda pushes you off the ledge again, the murderous broad. You now have to tilt the Wii remote to align yourself with the circle on the ground, and use the sailcloth to land softly.
The final cutscene of the intro finally sets the game into motion.