Learn To Corner - Take turns as tightly as possible to save ground. Some computer racers are not as adept at handling them, so cut inside your opponents to gain that extra edge.
Your Best Friend, The Motorcycle - The most maneuverable vehicle. As Gina says, if you master the crotch rocket, no one will be able to touch you. So practice, practice, practice. As they are similar in effect to the nitrous, you can use wheelies to pick up speed. Once you master the wheelie, it becomes an infinite boost.
Save Those Nitros Until The Final Stretch - Computer racers use their nitrous early. Hold yours in reserve until that final straightaway, and then blast by for the win.
Shortcuts - Find them. Use them. Win with them. Explore on your own in Cruise Mode, or go through a race slowly, and examine all of your options leading up to each checkpoint. Watch the computer racers. Sometimes they will tip you off to a ground-saving shortcut, as long as you don't mind smashing through a few glass windows.
Mini-Map - Your second best friend. Glance at it on the fly to plan your best route. Watch it for upcoming pink colored buildings; these will have shortcuts in them.
Battle Mode Powerups Special - Experiment with Pulse, Stealth and Stop to frustrate and bewilder your opponents.
Cut That Brake Cable - Some of the races you'll encounter will have several thousand sharp turns in them. Ease through the turns by tapping the gas (as opposed to full throttle); you will lose less speed than you would if you slammed on the brakes.
Know How Your Car Handles - The comparative handling ratings given to each vehicle are accurate, so pay attention to them.
Rearview Mirror - If you're in the lead, don't look back, because someone is gaining on you. Do look in your rearview mirror. You can see when someone is coming, and on which side they're on. A well-timed cross checking swerve will deter opponents from taking your hard-earned lead. Practice blocking opponents when they use SST and Nitrous.