By flapping at a specific point in the loftwing tutorial, you can break the normal sequence of text that the game tries to go through and cause the Loftwing tutorial to end 11 seconds sooner. The timing for this flap is right as you close the last textbox in Zelda's first sequence of text. If you are using button controls, the timing of the flap is more precise since the textbox has to be closed before the A button can be used to flap. After this point steer around a little bit, but don't perform another flap to keep the Loftwing Controls textbox on screen. Zelda will interrupt that textbox with her second series of textboxes. Make sure to cancel her text as fast as you can by mashing the B button. if you do it correctly, the first Loftwing Controls textbox will appear briefly after Zelda's second set of textboxes, before the Loftwing Controls Part 2 textbox takes it's place.
When you first fly to the surface, jump off the platform at the bottom of the lower academy stairs. Curve your skydive so that Link is aiming to the left and then call your loftwing. You will hit a spawn point closer to Faron than if you had just jumped off normally.
Link is actually granted the Spiral Charge as he finishes talking to Professor Owlan on Skyloft, right before jumping into the tutorial. You are not supposed to be able to get away from Owlan with the Spiral Charge however, as there is a ground trigger preventing you from escaping going into the tutorial. If you could jump over that trigger, you could clawshot up to the highest island on Skyloft and go to the Sky from there. This avoids the loading zone to the Spiral Charge Tutorial and leads you to the real Sky with the Spiral Charge. From there you can get the Pumpkin Soup and fight Bilocyte. This skips the tutorial completely.
There are 2 known methods to skip it.
Talk to Owlan and tell him you give up on learning the Spiral Charge. This removes the Spiral Charge ability. Enter a loading zone then go back to Skyloft and ask Owlan to start the tutorial. Instead of entering the Sky, die *off the edge of Skyloft** to respawn away from Owlan. From there clawshot up to the highest island and enter the Sky to skip the tutorial. You must die off the edge of Skyloft, otherwise you'll respawn next to Owlan.
By Skydiving down to the bottom of the waterfall cave, you can then enter the cave and use the Zelda & Loftwing amiibo to return to the surface. This is 12 seconds faster than the deathwarp method, however it is only possible with motion control diving.
After Zelda joins you in your quest to rescue your loftwing (after going through the waterfall cave), there is a small cutscene where Zelda sees and points at your Loftwing. That cutscene can be skipped with an incredibly precise jump-jumpslash. This is considered only worth it in a TAS.