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.
For items that can be held behind you, you can activate the item (like a banana) and hold it behind you while you pick up another item, thus enabling two items to be on hand at once (although they must be used in order of pick-up). Also note that green and red 3 shell groups work best for this since you don't have to hold to do this.
Press to hop. Hopping repeatedly is done to go around sharp turns. Hop turns are used occasionally to make slight direction adjustments. Hops are also the start of power slides. You can hop over gaps at corners to save distance. It is also possible to hop over opponents in the beginning of the race by getting a power boost and pressing immediately after. Don't make the mistake of hopping repeatedly on straightaways because it slows you down.
Press left or right and press and hold to start a left facing or right facing power slide, and then move the stick back in the opposite direction. Each time you move the control stick back and forth, the smoke will change color, starting from gray, then yellow, and finally to red. Slide around turns and corners as tightly as possible, unless specified otherwise.
Press and hold just before the light turns blue to get a speed boost at the beginning of a race. To save time, start pressing A before Lakitu starts counting down, and just before you would normally press , let go of for just long enough that you won't spin out, and then press again.
Press + and or while standing still to quickly spin around. This is only used in some races. You can also press diagonally down and left or right while spin turning to do a reverse spin turn, which is useful for some battle modes.
Press or to turn. Turning normally slows you down, so you should use power slides to go around turns so you can then release a mini-turbo. Turning without power sliding slows you down, so it should be minimized, but sometimes it is needed to make occasional slight direction adjustments when needed. When you do need to turn, you should aways tilt the kart.
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, unless you are doing the shortcut in Wario Stadium, in which case using Bowser can be faster because you can turn more quickly.