Press A to accelerate at the instant between the dimming of the second light and the flashing of the blue light at the start of the race. If you like Mario Kart 64 and I know I sure do, soon you should be able to know exactly what time to do this simply by the sound of your car. It sounds a little nuts, but it works. Use the above tactic first to learn how the boost works and then start to memorize the sound of the motor. A true master will have this tactic down in no time.
When something hits you, press A three times, but hold the button on the third time. If done correctly, your Kart will quickly boost up to its maximum speed.
For a better start to each lap in the Rainbow Track of the Special Cup, press R to jump slightly past the beginning of the downward hill. This way you will not slow down and will hit the ground running.
There is a shortcut on this track that takes you about half way across the board. When you start the race, stay to the right of the big turtle rock (to the left on mirrored courses). Drive to the left of the first jump and to the next three (right on mirrored courses). It is best to avoid all the jumps in the race since they slow you down and will be longer to drive. In front of the three jumps is a narrow jump. Ride this to the hole in the wall. You will need a mushroom (unless your either of the light characters) to make it all the way to the hole in the cliff. Drive through the tunnel and, at the end, turn towards the trees to get to the finish line.
Get up to top speed at the beginning of the track. Near the front, you will get to the top of a very steep hill. Stay all the way to the left and just when you get to the peak, quickly turn left and press R to jump. It is extremely difficult to accomplish, but If done correctly, you will land back on the track, half way through. This will bring you half a lap ahead of the other racers, which is quite an improvement on Rainbow Road.
There are two sets of speed tracks on a row on the Royal Raceway. The second one is a long distance jump which you can make every time. To add a little distance to it, press R to jump just before the edge. For a better recovery landing, without a slowing bounce, continue to hold R until you land. Your kart will impact hard, but you will be moving forward.
Note the first bump right after starting a race at Wario Stadium. When the race starts, turn around and go back a short distance. Next, accelerate to the left side of the bump. Press R a couple of times when you are at the top of the bump. If done correctly, you will land on the other side of the wall, cutting off half or more of the race.
If you have any speed boosting items when you get across the bridge you can give yourself a huge lead by not going on the road. Just hug the fence to the left of you the whole way around, you'll still cross the finish line and if you have enough of a boost you'll make it right back on the track.