To my knowledge (Assuming your playing the GC version) You hit one of the shoulder buttons (R I think, I havn't played in a while) and let go of 'A' because if you don't.... Then you won't. If you have Basic configuration on the gamecube then you press and hold (R-button) to drive and then for drift races the best thing to do is NEVER EVER let go of the accelerator, insted use the brake (L-button) and E-brake (A-button) to cut corners. keep goin' side to side on straightaways. as you approach a turn, let go of the gas hit E-brake real fast then floor it out of there but don't forget to keep "swinging" side to side to keep your drift goin'. remember you cannot keep a drift you started waaaay back there around the next corner, it's not possible. if you want to keep you drift points stop your drift before you get to the corner. trust me, it will save you alot grief, if you do what have said.
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It's easier with a RWD car, so start off by picking one. XRT is the most used, followed by the FZ5.
Then get a drift setup, or if you can't find one put the rear tyres with the maximum pressure, and the front ones with almost the minimum, that'll be a good start.
Then it's a matter of practicing and having good car control. A good track to practice is fernbay because it's full of curves for you to drift. Watch other people online and see how they do it, then imitate.
With practice you'll get at least as good!