Racing: Should be easy enough. The AI will be better no matter what you do and always catch up. However with boost drifting abuse and knowledge of the map you should be able to beat these.
Weapon battle: Very easy indeed. Simply stay behind the pack so you aren't bombarded with weapons and slowly pick off each AI one by one. Watch out for reverse shots of course and you should have this easy.
Drift challenge: Now this can be a little tricky but overall not too hard. You need to pick a character with the perfect balance. Knuckles fits this perfectly for he has superb drifting and his speed is perfect for the job. Make sure to learn where each checkpoint is as you don't want to use boost only to miss half the drift road because you couldn't control the car.
Versus battle: A very handy trick which will see you win every race with ease. Pick up weapons around the track until you earn either firework or snowball x3. Beat the very first opponent (you should have no trouble) and save the weapon. Now the next opponent will spawn. They will be invincible for roughly 3 seconds. Wait it out and unleash the attack. They won't be able to get away and it will be point black range. Use this to your advantage and go full race mode. They should fall into the red and be eliminated immediately within 10-20 seconds. Much better than driving for the whole 60 and guaranteed success.
Traffic: This is the ultimate toughie. For this I would highly advise either Sonic, Knuckles or Gum. You need a light car that can really drift with decent speed. This will be tough. The trick is to drift whenever possible as much as you can to get around cars quickly and boost away. Don't do it more than needed as you can crash and all that work can be undone in one crash and you could lose up to 6 seconds. Try to fool the blue police cars. Turn slightly left then immediately right. The car won't be quick enough to react most of the time. Watch the orange cars. They are easy to avoid but when you try dodging green and blue cars you tend to forget to watch where they moved last. Make sure to watch them. Finally and most importantly try not to miss the time bonus cars. These are absolutely vital for the last few waves. A trick I used so not to miss them is this: Check the mini map in the corner and you will see an icon for bonus time and next wave cars. Keep pausing the game as you get closer and look at the map to see where they are. I can't tell you how many times you will end up accidentally boosting past them. Even if you play like a god you NEED those time bonuses.
Finally ring races. I have no tips for these as they are so extremely easy even if you crash a few times you can pass with ease.