You can find eggs all over the four different planets here on res, go to hatchery and then to the creatu list to see every single kind of pet on here. When you find an egg ( random event anywhere on any of the four planets ), simply go to the hatchery and click incubate. The incubation minutes for every egg differs, the creatu list will tell you how long your egg will be incubating for. When it is close to 100% click hatch and BOOM! you have a pet. Give it a cute name and you're on your way. :D Pets also come in different colors, if you're lucky you can hatch a sepia, cream, ginger, silver, blonde or even an albino pet. Most of the time you will get naturals though. All of the other colors you will have to dye your pet to get, by purchasing a dye kit for millions of tu. Just take Whirlaway for example, when I hatched him he was sepia, now I dyed him pink. Though you cannot bean ( change the color to one of the natural ones ) or dye ( I think.... ) credit shop pets, which are the rarest of all of the pets.
Also, I suggest you try your shot at being a merchant, unless you're really good at naming pets and hatching colors :)
If you've got any more questions or things you would like me to clarify, please feel free to drop me a line via Rmail, welcome to Rescreatu - hope you have fun!