Taken from an awesome FAQ by Pete Philippis, you can see the rest at the following link:
Super Mario Kart Tips and Tricks
Practice firing green shells on turns at an angle that can send it straight down a corridor. Practice making turns without slowing down and hitting walls. The best way to make a turn is to hit the jump button coming up on a turn and then turn in that direction. Sometimes you may need to jump more than once. Practice lining the wall on the fly. When you get hit by a shell, hold the gas and hit the jump button. It seems hitting the jump button will break you out of dizzy spell quicker. After making a jump and turning, if you hold the jump button you will be at more of an angle. Try this on track 2. Start yourself in a circle, then jump and hold the button. After you release the button you'll notice your kart straighten up a little. Keep this in mind when you have a red shell and are being pursued. You can use this to your advantage to make a quick 180. Remember, you don't have to be fully turned around for your shell to turn around. It is possible to drop a shell or peel on top of a sitting opponent and other shells and peels. Remember, you can't spin-out when you are invincible. Learn to tell, by the music, when exactly invincibility will run out. As you near an opponent, if you shoot the shell just before you hit him, you and your shell will pass right through your opponent. If you are sitting right next to your opponent (whether sideways, behind, etc) and fire a shell, the shell will pass right through your opponent. Be careful. Firing a red shell like this will undoubtedly cause it to loop around and hit you! You are not immune to your own red shells that you have fired. If your opponent has fired a red shell and and is standing still or moving slowly, you might want to try to whip by your opponent and hope that as the red shell is coming around for a pass to hit you by following you, it takes the same path and hits your opponent. I usually have this happen once every couple rounds. Use the colors on the map to quickly locate your opponent and move in for the kill. Counterclockwise Red - Yellow - Green - Blue The manual that comes with the cartridge has a few useful hints listed on page 17,18, and 33 for battle mode tactics and tips. RACING TIPS ----------- Power slides slow your kart down. Rather, to do a more efficient turn, drive on the outside of the track and take the turn close on the inside (like they do in formula 1 racing). Perfect it so you don't slide or hit the wall. If you think you can't make it at the speed you are going, try letting go of the gas, or better yet, tap the gas which will slow the kart down yet still keep you at the fastest speed coming out of the turn. I recommend using Yoshi and Princess for the best results in race mode once you have learned the tracks. While Donkey Kong and Bowser have the best top speed, hitting other opponents and walls can be costly and you are bound to have it happen on the 100cc and 150cc races. Beginners should stick with Koopa and Toad because they are easiest to handle and can recover from crashes quickly. Practice, practice, practice. Learn where all the turns are so you can anticipate where your kart should be for best turns.