Opening lava gap skip
Skip the massive opening turn (always via the right path) by jumping the lava gap. Three itemless methods from easiest to hardest: the rail version with a wall clip off the black-and-yellow line, a coin-dependent charge jump up the wall into the same clip, or a raw left charge jump between the two bolts with an inconsistent wall clip. A mushroom or mega mushroom makes the charge-jump version 100% consistent, and a feather over the gap is the fastest and safest option of all - described as the most important shortcut in the game.
Post-cannon wall-jump turn skip
Trick early onto the wall, stay on it long enough for the trick boost to reach full speed (opposite of the first wall), and hold left toward the wall in the air; easy to fail without knowing these two rules.
Post-wall-ride mushroom cut (feather option)
Approach with a left drift and trick, mushroom through, and release the mini turbo halfway (holding for full orange ruins your exit alignment); with a feather this shortcut is completely free, saves over 3 seconds, and is nearly impossible to mess up.
Opening chain climb to upper level
From the starting line, aim at the giant chain leading to the top level, trick onto it and trick forward off it to reach the upper level with a guaranteed item box, grinding a couple of rails before dropping back to the main course. It doesn't save much in time trials but breaks you away from the pack in races.