Shortcuts are ways to help go throughout the course easier. Many are offroad ones, but some are more complicated. Many also require Mushrooms.
There is at least one in each race.
Luigi Circuit: Before the boost pads, there is a ramp where you can do a trick on.
Moo Moo Meadows: You can use mushrooms to speed over the grass, in the area where the cows are.
Mushroom Gorge: You can mushroom over the pit near the starting line, and you can also use a mushroom (the item) to ride on two brown mushrooms onto the bridge before the single mushroom.
Toad's Factory: You can mushroom past the moving platforms on the harbor.
Delfino Square: You can mushroom onto the dock a short ways before the bridge with the boost pads.
Rainbow Road: You can mushroom over the two large openings after the big hill.
Grumble Volcano: You can mushroom up to the rock with the huge box on it, and ride around it to count your laps. This is in several shortcut videos.