You can catch them out of the wild, you just have to use yours to beat it up and then throw one of the red and white balls to catch it in. After that it usually decides it likes you. Sometimes they decide they like you without any of that and they'll follow you until you agree to let them stick around.

You can also buy eggs from people who have extras, they get sold on the network all the time. Everything here hatches from eggs. I got a fox out of an egg, once.