In addition to knowing what to do in each course, it's also important to know why you're doing it, so you need a good understanding of the game physics. This will also help you pay attention to little details that you might not have thought about.
Basic Tactics
Weight Class
The lightweights, Toad, Peach, and Yoshi, are the fastest, have the best acceleration, and slow down the least on offroad surfaces. The three lightweights are identical to each other. Always use one of the lightweights in time trials, except in Wario Stadium shortcut fastlap where you should use Bowser. Bowser is the only heavyweight with fast acceleration ability.
Rocket Boost
Press and hold A just before the light turns blue to get a speed boost at the beginning of a race. To save an extra .01 seconds, start pressing A before Lakitu starts counting down, and just before you would normally press A, let go of A for just long enough that you won't spin out, and then press A again.