You can find animals for your Chao hidden across the different levels. This page will list all the animals are found in which levels. You can find these animals inside the robots created by Dr. Eggman, roaming around the level, and also hidden in boxes, cages and pipes. You can only hold up to ten animals at a time, which you can see displayed at the bottom of the screen.
Here you can see which levels contain which animals. The first four animals can be found within Dr.Eggman enemies, but also in unusual places - cages, hidden inside pipes, or inside the second Chao Container.
In order to destroy cages, you should look for a rocket nearby which you can fire by pressing a button. In order to get an animal out of a pipe, you need to stand next to it and whistle with the action button. You can also find these animals inside the robots created by Dr. Eggman, such as the bomb-throwing monkeys, or the robots that look like E-102 Gamma — please note that the majority of robots found in the game are G.U.N. robots, which contain Chaos Drives instead of small animals.
The fifth animal listed for each level belongs to either the gold or black groups, and is found in the third Chao Container; it can also sometimes be found in very rare locations in the level.