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).
Unlike all other Chao games, the Chao's face on Sonic Adventure 2 Dreamcast seems to be chosen at random when the egg hatches, and not set when you first enter the garden. Youtube video coming soon!
Please note that while many people believe otherwise, the way you hatch the egg does not seem to have an effect on the Chao's facial appearance.