Mario Kart 64

Shortcuts in Koopa Beach:
Probably the most famous and well known and executed Mario Kart Shortcuts, the legendary Waterfall Shortcut. After you down under the arch about halfway down the track you'll see a ramp near the rock wall, and a hole in the wall. Take the ramp and enter the hole into a tunnel and exit further on in the track, you'll come out of a waterfall.
It's easiest to do this with a Mushroom, but can be done without if your going fast and jump just at the end of the ramp.
(It's also funny to shoot a red shell at your opponent just as he/she is going up the ramp, lol.)

There is also a second little shortcut in Koopa Beach. Instead of going under the arch, simply go over the right of it, there is a small slit of ground going forward, take it and cut a few seconds off your race.

Shortcut in Rainbow Road:
My brother loves this shortcut, takes it every chance he gets.
If you have a Mushroom at the start of the track when you going down the hill use the mushroom and jump off the course to the left, if you time it just right you'll land on part of the track much further. But it easy to just miss and just fall until the Lakitu picks you up and puts you back on the track.
I've also seen this work if you get the speed burst at the start of the game.

Shortcut in Wario Stadium:
This is a hard one, but right at the start when you're bouncing off those little hill-bumps, you can jump over the wall and take off alot of time from your race, this is pretty hard to do though, and can cost you time if it doesn't work.

Speed Boost at Start:
Press the A Button right as the last light is fading away.

Princess Peach's Castle:
You can drive outside of Princess Peach's Castle in Royal Raceway, right after you take the big jump over the river, you can turn right and be there. Doesn't do anything, it's just for fun. Pretty cool to drive around there if you've played Super Mario 64 a bunch of times.

Lightening Trick:
This is something you can do on a few tracks, and it'll bug your friends ALOT!
If you get a lightening bolt, in Wario Stadium for example, right before that jump near the end use your lightening bolt, they'll shrink down and spin out, they won't make the jump, but fall down and be behind in the track.
This can be done on other tracks such as Royal Raceway before the big jump.
Me and my Bro got so mad at eachother for doing this so many times over the years we finally agreed to not do it to eachother.

Koopa Beach Shortcut Trick:
This is another annoying thing you can do to friend when your taking that ramp in Koopa Beach to the tunnel. If you get a Banana, a trail of Bananas, or a "?", take the ramp and just as your going up it place it on the ramp. If your opponent is right behind you he can't take the shortcut.
Perfect for final lap.
But again, will bug your friend, and can be considered a "cheap trick."