Okay I won't list every shortcut in the game. The ones where you cut through places are "obvious". Here it goes...
Peach Beach: At the first part of the beach area with the cataquacks, go straight instead of right. You'll come across a ramp with a turbo. Only come here with a mushroom, because if the tide is high enough, you could get slowed down from this shortcut!
Mushroom Bridge: After you exit the 1st tunnel, go straight (Make sure you have a mushroom when you do it!) You will go up and down a ramp.
When you reach the bridge, go up the sides of it (IF you're good at holding your balance). This isn't really a shortcut but there are a lot of turbos on the sides of the bridge.
Mushroom CIty: In one part of the city, you can turn right to go on a path between buildings.
Later in the level, towards the end, you can use a mushroom to get across the right part of the area.
Yoshi Circuit: Before one of the big turns, you can use a mushroom to jump over the pond and skip the big turn.
Later in the level, you can use a starman to go left and to a tunnel shortcut. Be careful--it's really hard to reach and you may end up falling into the water!
Dino Dino jungle: At the part where you reach the bridge that splits in 2 paths, there's actually a 3rd one that takes you through it faster, but make sure you're good at holding your balance or you'll fall into the water.
Once you reach the cave after that, you can go right using a mushroom.
Bowser's castle: In the room with the firebreathing Bowser statue, you can jump over one of the holes. You can even do it without a mushroom, but don't take it if the Bowser statue is breathing fireballs or you'll fall into the lava!
That's it. I haven't played this game in a while, so my shortcuts probably weren't accurate.