There are a number of emblems to be gained by clearing the Chao Races, so it's worth doing, if only to unlock more cool stuff for your Chao!
When you first enter the waterfall door, only the Beginner races are open to you.
Pick a race
Beginner|Jewel|Challenge|Hero|Dark
This is seriously easy. There are four courses: Crab Pool (swimming), Stump Valley (flying), Mushroom Forest (running) and Block Canyon (power).
Crab Pool: All you need is a Chao with around level 10-15 in Swim and a similar amount of Stamina, and you'll win all three levels easily. The course is just to swim across the pool. Once your Chao has cleared all the levels, he'll get the Shovel as a prize, and a medal to wear on his chest.
Stump Valley: Another very easy course - your Chao just has to fly across one gap. Give him 10-15 levels of Fly, a similar amount of Stamina, and perhaps a couple of levels of Power; if he doesn't make it across the gap he'll have to climb up. The prize is the Watering Can and a medal.
Mushroom Forest: A straight sprint to the finish. Any Chao with level 15 Run and Stamina can beat it without a problem. The prize is a Rattle and a medal.
Block Canyon: Have 15 levels of Power and 10 of Run, and definitely have 15 or possibly more levels of Stamina. Your opponents on level three are a bit tougher than usual. The prize is a Toy Car and a medal.
Now we're getting to the good stuff. The Jewel Races are accessed when you beat all the Beginner Races. Now you're racing fully grown Chao instead of babies. Each Jewel race has five levels of difficulty, and you'll need a Chao with good stats to beat the final one! Try not to use Chao with Ds and Es here; you want all Cs or better, ideally.
Oh, and while I don't mention Run, always have at least 30 levels of Run on a competition Chao. Even for Topaz, being able to move fast on the flat can make a lot of difference in a close race. It'll give you a head start, or a boost at the end.
Aquamarine: The Swimming cup, and you're competing against a whole spectrum of Neutral Swim Chao. Only enter Chao with level 50 or more of Swim and Fly and preferably level 50 or more of Stamina. If your Chao wins level five, he gets a Sonic Doll and an Aquamarine Badge.
Topaz: The Flying cup, and you're competing against a whole spectrum of Neutral Fly Chao. Only enter Chao with level 50 or more of Fly, some Swim, and preferably level 50 or more of Stamina. If your Chao wins level five, he gets a Broom and a Topaz Badge.
Peridot: The Running cup is a straight sprint to the finish against all kinds of Neutral Run Chao. Pour everything into Run for this one, with a good amount of Stamina - it's a long one and you don't want your Chao to run out of juice. There's a Run/Power Chao in level five who's a real bugger, so level 60 of Run or more.
Garnet: The Power cup tests strength and speed, so have Power at a good level, 50 or 55. Definitely have a good level of stamina here, even if you skimped in other races.
Once you've beaten the four Jewel Races, two more become accessible. Onyx tests Luck and Intelligence, and Diamond tests everything. This is where things get rough.
Onyx: Kindergarten lessons are said to raise Intelligence, as well as the Mushroom from the Black Market. Give your Chao plenty of both, and get all his stats to level 60 or more, including Stamina. If your Chao survives level five he gets a Colouring Book and the Onyx Badge.
Diamond: This is the big one - if you beat the Diamond race, you are the champion! Only enter Chao with decent grades and all stats level 60+. The course is very long, so pay extra attention to Stamina. If your Chao wins, he gets the Bubble Blower and the Diamond Badge.
These are unlocked when you beat the Beginner Races; you'll face a series of unique opponents. The prizes at stake are some awesome toys for the Neutral Garden that all the Chao (not just the winner) can use.
Level 1 - The Strong Gold and Silver Rivals: They're not that strong. Level 25-30 on all stats and you'll beat these metallics easily.
Level 2 - He Returns: Aaah! Oh no! It's CHACLON from Sonic Adventure 1! Thankfully he seems to be a bit out of condition... he's pretty easy to beat now. If your stats are all level 30-35 you should have no trouble.
Level 3 - I'm Trying: Heh... it's an army of Omochao. Anyway, he's a wuss even when there are seven of him; if you beat Chaclon before, you can beat this race.
Level 4 - Small Animal Challenge: This is a peculiar race against the animals themselves. Have all stats about level 45-50 - they're not as easy as they look. If you win, you get an awesome bouncy ball for the Neutral Garden.
Level 2-1 - Cockroach Chao: Ew. Level 50 on all stats and you'll crush these guys like the bugs they are.
Level 2-2 - Oh My Egg: You're racing against eggs now. Being eggs, they're not particularly good. If you beat the Cockroach Chao, just go straight into this race and you'll probably win.
Level 2-3 - Be Afraid! Skeleton Group: A bunch of Chao wearing skulls on their heads. Though cool, they're not particularly good, so again, if you beat the eggs, go straight into this one.
Level 2-4 - Be Afraid! Pumpkin Group: A bunch of Chao wearing pumpkins on their heads. You probably will need to level up a little for these guys - try level 55-60 on all stats. Beat them and you get a Jack-in-the-box for the Neutral Garden, so your Chao can practice their l33t Jack-in-the-box opening skills for the Onyx course.
Level 3-1 - Be Afraid! Ghost Group: These guys are not easy - after all it's the final tier now. I recommend level 65-70 to beat them.
Level 3-2 - He Strikes Back: Oh no... Chaclon has a score to settle, and this time he's more like his Dreamcast self. He's all right if you're at level 70 on all stats.
Level 3-3 - Chao Rangers: A veritable rainbow of shiny Black Market Chao. They look awesome, and they are pretty good, but not an immense challenge. If you beat Chaclon's second challenge without trouble, they'll fall before you.
Level 3-4 - The Ultimate Rival: This is it! You only have one opponent, but he is the Light Chaos Chao! This is the toughest race of all; have all stats level 80+, and the higher they are the better. He seems to have much more luck than your Chao: you will trip, you will muff the jack-in-the-box, you will fall down a trapdoor on the final straight, etc. With a very good Run stat you can make up the difference. The prize is the television for the Neutral Garden, and a nice looking medal your Chao can show off to his friends.
These are accessed along with the Hero and Dark Gardens when you evolve a Chao to be Hero or Dark. A Hero Chao cannot compete in the Dark Races, and vice versa. Neutral Chao can enter both Hero and Dark races. There are four levels of the race against unique opponents, and winning gets you special items for all the Chao.
Level 1 - Baby Devil: Your Chao faces a bunch of young Dark Chao. If all your stats are at level 25-30 it's a breeze. No prize yet.
Level 2 - Child Devil: Actually, these are evolved Dark Chao, not children, but eh. They're a lot tougher than the babies, so have your stats at around level 40. Winning gets you a cool ball for the Hero Garden.
Level 3 - Death Troops: A bunch of Dark Chao wearing skulls on their heads. Despite their goofy looks they are hard, so have stats at level 55-60 with extra Run. No prize for this one...
Level 4 - Devil Versus Angel: This is it! You only have one opponent, but he is the Dark Chaos Chao! This is the toughest race of all; have all stats level 70+, and the higher they are the better. He seems to have much more luck than your Chao: you will trip, you will muff the jack-in-the-box, you will fall down a trapdoor on the final straight, etc. With a very good Run stat you can make up the difference. The prize is a rocking horse for the Hero Garden, and an awesome looking medal.
Level 1 - Baby Angel: Your Chao faces a bunch of young Hero Chao. If all your stats are at level 25-30 it's a breeze. No prize yet.
Level 2 - Young Angels: Have your stats at around level 35-40 to beat this cute bunch of evolved Hero Chao. Winning gets you a cool ball for the Dark Garden.
Level 3 - Flying Angels: Fly and Swim are the main stats here. Have them at around level 55, and make sure your Run and Power are at least level 40. As always, have good stamina, and you should win.
Level 4 - Devil Versus Angel: This is it! You only have one opponent, but he is the Hero Chaos Chao! This is the toughest race of all; have all stats level 70+, and the higher they are the better. He seems to have much more luck than your Chao: you will trip, you will muff the jack-in-the-box, you will fall down a trapdoor on the final straight, etc. With a very good Run stat you can make up the difference. The prize is a radio for the Dark Garden, and an awesome looking medal.