When you fall off the track, when the little cloud man bring you back to the track, if you press the gas button right when the wheels touch the ground, you'll also get a little boost. Courtesy of fUble_baSs.

You get a boost at the start if you press A right after the lights turn green. You can also get a much larger boost though in co-op when both players press A at the same time. Courtesy of Duncan.

To receive a boost at the beginning of a race, hit A at the same time the announcer says go. If timed correctly, you will jump to the head of the pack.