With this trick, you can dive in certain areas without even having beaten the water training barrel.

Drop into a body of water that is slightly moving up and down. When the surface is at its lowest point or close to it, press C-up. This will keep you in position as the water is moving up again, letting you go beneath the surface. As soon as you see the screen turn bluish, go back to third person view and start diving.

In fact, the only thing disabled until you beat the water barrel is the Z trigger on the surface. Therefore, another alternate way to start diving is to enter a body of water below the surface from out of bounds.