I know a lot of shortcuts so I'll just post my two favourites.
Sky Garden Mushroom Required: Yes, I can do without, but it takes a lot of practice
Time Saved: about 5 seconds
Difficulty: 2/10 with mushroom 7/10 without mushroom
When you reach the second coloured ramp (has item box above it in grand prix) go to the left and keep going straight, use a mushroom to jump across the gap.
Rainbow Road Mushroom Required: Yes, can be done without, but insanely hard
Time Saved: 2.5 seconds
Difficulty: 10/10 with mushroom 11/10 without mushroom
At the end of the spiral at the start you will find your self at the top of a hill. From the top of the hill (from left of it too) use a mushroom and tap R to jump. Direct you're Kart towards the right side of the loop so that you land on the rail, you must fastfall (press up on the d-pad). Your Kart will be riding on the rail, jump off to the right after about half a second onto and uphill section shortly after the loop. If you manage it, you're amazing, I havn't managed it yet, but I've gotten close.