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 start, you will have two Chao eggs in the Chao Garden. As you later unlock other Chao Gardens, you will find that each one of those has a single Chao egg. These eggs contain regular, normal coloured Chao, which are light blue with yellow highlights. They will change in shape and colour as they grow.
There are three ways of hatching Chao eggs, with each way having different effects on your Chao.
Shaking or rocking the egg is the most common way of hatching an egg. Pick up the egg using the B button, 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 by pressing the B button, and then throw it by pressing the B button again while moving. If you throw the egg into open space it will not break open, you will need to throw it against a wall.
Your Chao will not look very pleased at first if you use this method, but there do not seem to be any negative effects from doing this. All that will happen is that the Chao will start with an angry/upset expression above its head; however, this expression will only last a few moments, and the Chao seems perfectly happy afterwards.
Many people choose to hatch their eggs in this way because it is far quicker than the other two methods. Some guides, manuals, and magazines have said that this method produces a more active, lively Chao, but this is unconfirmed.
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.