There's also a sidequest in there that allows your Chao to gain the Morale Meter, which can give your Chao temporary stat boosts. To gain it, after you have gotten the Engine Core for the first time, go to this stage in Trial Mode and destroy all of the soldiers in the stage to recieve a large point bonus. The Colonel of the soldiers will pop up. You should be able to easily destroy him with a few well timed homing attacks. He will drop a key of some sort. Walk around the room with the Engine Core to find a secret room. Get the item and the stage will end, only now you will be transported to the Egg Carrier Chao Room. You now permanently have the Morale Meter for all of your Chao. I'll explain directly how it works, so you can use it to your advantage:
When you pet a chao or carry it around, as you know, it develops a relationship with you. The more you raise and show your affection for your Chao, the more it will be willing to do things for you; in this case, win the race! Developing a positive relationship with your Chao can increase the charge-up time and the capacity of the Morale Meter.
Now, here's the important facts you'll need about what to do with the Morale Meter. When a race starts, your Morale Meter will begin to fill. When you wait long enough, it will gain enough power to give your chao a powerful yet temporary boost to all stats. Note that the meter fills faster if you are in 1st Place, because your Chao's pride will fuel its morale and thus increase the effect of the Morale Meter. Also note that it can only be used once per race, so charge up and let loose when needed!
I hope this helped. I know you aren't exactly the champion of the games on TSC, and I just wanted to help, because all of the best racers know how to use upgrades to their advantages, and unlike some players here, I'm actually willing to give something to the people. :)