Hey Shiori! Welcome to Res. :)
There's a lot of things I could explain, but since you don't seem to have a pet yet, I'll start there. :) You find eggs in random events while exploring the four planets. It will say "While walking around you have found a __ creatu egg. Pick it up.", and if you click on "Pick it up", you will get the egg. Then you can incubate it (takes a different amount of time per egg type), and hatch it. :)
You can see the different types of pets, which world you find them on, their rarity and incubation times in the Pet Directory (Go to "Information" in the menu bar and then "Creatu Species"). Each creatu can hatch into one of 9 colours, and sometimes a 10th colour (achromatic).
You can hatch 3 eggs a day, and you can earn a 4th hatch through the daily rewards system. Your pets will naturally grow through 3 stages over 30 days, but don't forget to feed them, or they might die.
Hope this helps. If you have any questions, or want to know about anything in particular, or if you want some help finding a particular type of egg, post here or send me an rmail. :)