Adult chao are able to breed, which basically means you can choose to make a baby chao from mating just about any two adult chao. First of all, to get your chao to mate with another chao (to make a chao egg), you must wait for your chao to be in mating season, or you can go to the black market and buy heart fruit. If the heart fruit isn't there, you either should try beating more missions (you shouldn't have to complete that many), or just keep coming back periodically to see if it's for sale. But anyway, once you have bought the heart fruit or sat and waited until mating season, you should see that either way (or after they ate the heart fruit), they'll have a circle of flowers growing around them. Sit your two chao near eachother while this is occuring, and eventually they should go close together and start making a kissing motion. About 5 to 6 seconds later after this motion they'll both jump backwards and an egg will come right out of the ground. This chao egg once hatched as you see will contain special kind of identities from it's parents, which brings us to a whole new route, chao genetics (genome).
Here's an important fact about breeding. Character and Chaos Chao are not able to enter mating season, meaning they will never have the flowers grow around them during mating season or after being fed heart fruit. Baby chao also can not mate, only adult chao. Inorder for your chao to be an adult chao, it must have evolved once (example, into it's hero, neutral, or dark state). The products from two chao mating will always be an egg containing a baby chao.