Hey there, Kara! To get a pet, you have two (?) options:

Buy an already hatched one from someone's rancher shop, or search the planets via the explore page and wait for an egg to pop up. Once you have an egg, go to the hatchery page, incubate it and wait until it reaches a hundred percent (On the creatu list, it tells you how long this takes. You can hatch it early, but then you run the risk of losing your pet). Once it hatches, name it and there you go, a pet!

Does that help? C: