To gain access to your Chao, you must first find the Chao Garden. There are three Chao Gardens that you can find located in Sonic Adventure; one in each adventure field.
You can start raising your Chao once you have discovered and entered a Chao Garden. Time only passes for the Chao while you are inside the Chao Garden. If you are exploring an adventure field, playing a stage, or are in a different Chao Garden to the one your Chao is in, it will not age.
Your First Chao
When you first enter the Chao Garden, you will see two Chao eggs. As you find the other Chao Gardens, you will find two Chao eggs in them too. These eggs contain regular, normal coloured Chao, which are light blue with green highlights. They will change in shape and colour as they grow.
There are three ways of hatching Chao eggs, with each way supposedly having different effects on your Chao. It is recommended to feed your Chao once it is hatched to increase its happiness!
Shaking or rocking the egg is the most common way of hatching an egg. Pick up the egg, and while still holding the button down, move the analogue stick to shake it. After about 5-10 seconds of shaking the egg, place it back down on the ground. The egg will start to bobble, and then your Chao will hatch from it.
You can also throw the egg to break it open. To do this, pick up the egg and then throw it by pressing the B or Y button again while moving. You Chao will initially not be happy, will cry, and will even hiss at you if you come near, but it will not lose any HP. Cheer it up by feeding it!
The help orb in the garden will tell you that the way you hatch the egg affects it's personality. A lot of sources say that this method produces a more active, lively Chao. I have put a newly hatched chao (via throwing) into a VMU and it's status was given as "Lively". However it is unconfimred what difference this makes to the Chao's behaviour in the Chao Garden.
Your Chao egg will hatch naturally after a while, even with no interaction from you. Simply leave the egg alone and it will hatch by itself after around five minutes. Hatching a Chao this way has been said to produce a normal, perhaps timid Chao; however, as above this is unconfirmed what differences this makes to the Chao's behaviour in the garden.