Hold the gas while tapping the brakes to get your tires smoking. When the countdown is between '1' and 'Go', release the gas, then press it again.

When you are about to start a race, hold X during the countdown and tap Square. This will make your tires spin and you will gain speed faster.

As soon as the race starts and the countdown begins, hold X + Square. As soon as the countdown reaches "1", release Square. To do this correctly, you must have some sense of rhythm. Wait until the countdown is three quarters of the way down between "1" and "Go", then release X and quickly press it again. If done correctly, you will get a boost start.