Instructions
Go to the Chao Garden and find a Chao egg to make it your own. It will hatch either if you wait for a few minutes, or if you pick it up and rock it. After it hatches, you begin taking care of it. It will give you visual cues to what it needs. The Chao Garden is located in the Hotel Lobby. There are two elevators, and entering the elevator on the left will take you to the Chao Garden.
Feed your Chao fruit. If you want your Chao to be as healthy and happy as possible, feed it as often as possible without overfeeding it. But if you're only concerned with getting the second evolution of the Chao, and not its happiness, go ahead and overfeed it. When your Chao is overfed, it will cry. Getting a feel for how much to feed it before it cries takes a bit of time and observation.
Focus on the Chao you want to evolve, taking care of it as frequently as possible. Most of the Chao's evolution is a waiting game, but the more effort you put in, the shorter the wait becomes. A Chao year is only a few in-game hours, and their lifespan is about five years. So a little effort will yield plenty of results.