You MUST have a working VMU in order to check the stats of your Chao.
Every so often, place the Chao in the VMU and check his stats. 999 is his/her max.

Also, you have to give the Chao the animals in this order. Here's why:
The Cheetah will lower flying and swimming. The Seal lowers flying. The bat lowers nothing. And the Dragon lowers nothing. If you work on swiming first, then on running, your chao will actually get slower.

You'll basically need only 3 types of animals. Cheetah, Seal and Bat.
Note: This stratagy guide is based around a normal chao, who's stats all raise at a normal rate.

First we will work on speed. For this, we call for the cheetah.
The cheetah is ideal for raising speed and easy to find. Use Knuckles and go to Wild Canyon. Just go to the statue in the corner to the left at the other end of the room and you'll see a cheetah. Collect it and restart the level to get another one. Repeat until you have 10, then go to the Chao Garden and give them to your Chao. After the 10 cheetahs, you'll notice running stats increase by 40. Repeat this process until you reach 999 for running

Running low on lives?
Well, just simply reload your file after you exit the garden (make sure you use the transporter in the main area of Chao World, so the Garden data saves). As long as you don't finish any levels, when you reload, you're lives will be back to where they started.

Okay, next step is Swimming. For this, I call apon the mighty seal! heh.
Anyway, this one raises swimming just as much as the cheetah raised running. This little guy is easy to find as well. Go to Aquatic Mines as Knuckles and jumps in the water to the right of the platform in front of you. Swim down and collect the seal at the bottem. Like with the cheetah, restart the level and repeat until you have 10, then return to your chao and give him the seals.

Right then and there, you're Chao should be good to win ANY race, regardless of Flying being 0. You will notice, however, that power has increased quite nicely. This is because the Cheetah and Seal raise power a little too. But to raise a true super Chao, we can't stop now! ^^

Now it's time to work on flying!
For this, the ideal creature is the condor, but the condor will GREATLY lower your other stats.. so lets use the next best thing, the bat. The bat wil raise flying by as many points as the seal raised swimming. The bat is easily found in Pumpkin Hill and Pyramid Cave. I suggest Pumpkin Hill. There is a bat by one of the train rails on the side of Pumpkin Mountain. As usual, collect, restart, get 10, give to Chao.

Okay. So now we've maxed Run, Fly and Swim.
Is power maxed yet? If not, try the dragon. The Dragon can be found easily in Wild Canyon. It will be in the 3rd Chao Box. The boxes are easy to find, as this is a small level. In box one, collect the Chao Key. Box twi is full of animals, just break the box, don't collect the animals (this leaves more inventory room for Dragons).
You know the drill after collecting the dragon right? Well there's 1 difference this time. DON'T restart. Instead, exit the level and reload. When you restart, the animals and keys in the Chao boxes won't be there again sicne you already broke them. Besides this oen difference, the rest is the same. Collect 10, give to Chao.

Just keep giving your Chao all these animals until stats max out and you're Chao will be unstoppable!

Okay. You must be saying "What about Luck? Brains? Stamina?" Well, with all you're other stats maxed, you don't need these.. but like I said before, what's a Super Chao without them? ^^

Unfortunatly, I have no sure fire method for these stats.

Luck is only raised when playing Chao Adventure 2 in the VMU. Luck raises by 100 when you find some boxs and pick the correct one to open. Also, luck lowers by 100 if you choose wrong.

Brains is raised by going to the Chao Kidnergarden and by eatting the Wisdom fruit you find in Chao Adventure 2.

Stamina is raised whenever your Chao eats something. Feed him lots of Chao nuts to raise stamina. He/she has to eat the whole nut to gain any stamina, so if he/she tosses it away before finishing, pick it up and give it to him/her again.

I used this method myself after my old Chao reached the part of his life span where he reverted back into an egg and his stats reset. I was so mad, because I only had one Chao race left to beat. Using this method, I raised his running to 999 before he even matured!