Get a speed burst at the start of a race:
Press the A button just as the last light is fading away.

Shortcut in Rainbow Road:
Get a Mushroom for Speed. At the very start of the track when you start going down that steep hill, jump and turn to the left. Hopefully you'll land on a part of the track later on in the race. You can do this with the speed burst you get at the start, and if I remember correctly, it is even possible without a speed burst.
The first time you may miss, but after a few tries, you should get a pretty good idea of how to do it.

Shortcuts in Koopa Beach:
The most famous shortcut I'd say. After you drive under that rock arch halfway down the track you'll notice some ramps. The last one is smaller and if you use it properly, you'll enter a tunnel, exit through the waterfall later down the track. It's alot easier if you use a mushroom or have a star, but its possible without one, just drive straight at it with speed, and jump right before the end.

Shortcut 2 is less taken, instead of going under that arch, just go to the right. There's a small line of land, which can cut a few seconds off your race.

Wario Stadium Shortcut:
This is probably the hardest one to pull off. Right at the start when your going off those little bumps, try and jump over the wall, if you make it you'll cut alot of time off your race. But it is harder, and can slow you down a bit if you miss.

Lightning Trick:
If you've played the game enough you've probably done this. But there are crucial points in certain tracks where if you use your lightning bolt you'll shrink your opponent and make them mess up alot.
In Wario Stadium, if use near the end when you take that jump, if used before they'll fall off, to a part further back.
In Royal Raceway, if used a little bit before that big jump, they'll fall off into the water.
In Koopa Beach, right before they try and take that jump.
There are tons more, just use it at the right time.

Princess Peach's Castle:
Just a cool little extra. In Royal Raceway after you take that big jump, you can turn right, and enter the area outside Peach's Castle. It's quite a cool thing to see, especially if your a fan of Super Mario 64.