Crazy Taxi is kind of a tough game. Mostly because if you play the arcade game, you have no indication of how to perform the advanced techniques that you need to know to be successful at the game.
But that's what I'm here for.
You need to know 4 techniques to be a successful cabbie.
1. The Crazy Dash - The idea is to shift into Drive and then immediately stomp on the gas. The timing is a little tough to get without a demonstration, but pulled off successfully, you'll immediately rocket to top speed. You can do this at any time, even while you're driving. Also, if you're at a stop, and you do the Crazy Dash, but immediately put the car into Reverse, you'll go full-speed backwards.
2. The Limiter Cut - Once you get going at top speed with the Crazy Dash, you'll notice that your car will start to vibrate. When that happens you want to throw the car into Reverse, then Drive, and then hit the Gas, with the same timing as the Crazy Dash. This will give you a burst of speed that lets you go faster than your top speed for a couple of seconds.
3. The Crazy Drift - While you're driving, put the car into Reverse and then into Drive while turning a direction. You'll easily slide around corners. Let go of everything and do a Crazy Dash to recover.
4. The Crazy Stop - Do the Crazy Dash, but hit Reverse and the Brake instead of Drive and the Gas. You'll manage to stop much faster.
Here's a video I prepared of me performing all 4 of these techniques on the Dreamcast version Crazy Box challenge rank S-S. I'm a little out of practice at the game, but I think you'll get the idea.