Every Chao begins its life in an egg. Eggs come in all kinds of colours and patterns, indicating the breed of Chao that will hatch from it. Every single breed has a corresponding egg to go with it. Unfortunately, most of these are unobtainable without hacking. This is because most breeds must be created through mating, and mating always produces a normal-looking egg; therefore, the eggs for those breeds can normally never be seen.
The kinds of egg that are available without hacking are the Two-Tone Normal egg, all Mono-Tone colour eggs apart from Mono-Tone Normal, and all Jewel eggs. The Mono-Tone eggs can be purchased from the Black Market. Two of the Jewel eggs can be found in the Adventure Fields (see below); the rest of the Jewel eggs can only be purchased from the Tiny Chao Garden. The Two-Tone Normal eggs are what you start off with, but more can be bought from the Tiny Chao Garden for free.
When a Chao hatches from its egg, two halves of the eggshell will be left. One of these cannot be interacted with and will simply disappear when you leave the Chao Garden. The other piece will stay in your Chao Garden, and can be picked up and thrown around. It is actually a hat! If your Chao is able to wear hats, then you can give the eggshell to it and it will put it on. See the Hats page for more information. You can also simply take the eggshell to the Black Market and sell it; this way, every Chao that you hatch earns you some rings!
As mentioned above, two of the Jewel eggs can be found in the Adventure Fields. These are the Gold and Silver eggs, and they are two out of three secret eggs. The third hidden egg is just a Mono-Tone Black egg, which can already be purchased from the Black Market (albeit for 2,000 rings).
Each of the three hidden eggs is in a different Adventure Field. The instructions to get them are below. Please note that you can only find one of each egg per save file! If you have already taken the Gold egg, for example, then you won't be able to get another one on that save file.
The Gold Egg in Station Square
This egg can be found with any character. First, go to Station Square, and head to the area with all the shops, where Sonic first fights Chaos 0. There is an antique shop nearby. If you look through the window, you can see a shiny stone on display on a pedastal. This is the Gold Egg! However, if you walk inside and try to pick it up, you will see that it is not so easy to take. When you pick up the egg, the shop's alarms go off and you get locked inside!
To take the egg, you will need to leave the shop and move to the enclosed area by the town hall. There is an egg-shaped rock hidden in a grassy corner of this enclosure. Pick up the rock and take it back to the antique shop. Then, simply replace the Gold Egg with the stone on the pedastal; this will trick the shop's alarm! You can then carry the Gold Egg back to a Chao Garden.
The Silver Egg in the Mystic Ruins
This egg can be found with any character. First, go to the Mystic Ruins, and head to the lake which is opposite Tails' house. You will see a waterfall, and if look closely, the Silver Egg is there behind it! However, it is locked into place by some strange mechanism.
To release the egg, you need to move over to the right side of the lake. There is a large button built into the wall, which looks like a small platform. Just walk into the button to push it. A few seconds later, the Silver Egg will be ejected into the lake! Simply wait for it to float ashore, and then pick it up and take it to a Chao Garden.
The Mono-Tone Black Egg in the Egg Carrier
This egg can only be found by Amy or E-102 Gamma. Using one of those two characters, head to the Egg Carrier. Go inside the main interior chamber, which is very thin and long. One end of the room has the entrance to the Egg Carrier Garden, with buttons in front of it to spell out EGGMAN. Go to the opposite end from here, where there are three doors. These doors lead to the Hot Shelter and the hammer mini-game.
Go through the right door, to the hammer mini-game room. If playing as Amy, you will need to hit the small button outside the door with her hammer, in order to unlock it. Then, continue through to the door on the opposite side of the hammer game. This will take you to the prison, where Amy was being held captive during her adventure. You'll find the Mono-Tone Black Egg in one of the cells. Simply jump on the button outside the cell to open the door. You can then pick up the egg and carry it to a Chao Garden!