On Koopa Troopa Beach, there is a shortcut in the first wide open area. There is a ramp leading to a gap in one of the rocks. You'll need a Mushroom or a Star to be able to jump the distance, but you can cut through over a third of the track by going through here.
On the Kalimari Desert level, it's possible to break away from the rubber band AI and they won't be able to catch up with you. To do this, on the first lap you'll need to get over the first train track just as the train passes. The CPUs will just stop. Continue on to the next train track. Try to get over it before the next train reaches it crosing, which you'll probably need a Mushroom or two for.
If you execute this quickly enough, the AI will lose its' rubber band effect and your two rivals will not be able to catch up to you.