Your Chao can have a variety of different faces. You can see the different eyes and mouths possible below. Click on the links to update the image.
Your Chao will start with an entirely random face, and it is impossible to change the Chao's face during its life, although it can change through reincarnation. The face that it gets seems to be influenced by the faces of its parents (if the egg was recieved via mating) and also strongly influenced by the face of its previous life (if egg was from a reincarnated Chao).
The Chao's face on Dreamcast Sonic Adventure is determined when you first enter the garden, and not when it hatches.
Please note that while many people believe otherwise, it has been proven that the way you hatch the egg has no effect on the Chao's facial appearance.