To adopt a pet on Subeta you can either create a new one or get one out of the pound. Certain pets like the Cadogre and Endeavor are released in limited number by Calvin Blackmoon at his workshop. The Bumbus, while not being limited, is only available from the Blue Building.
To create a new pet simply click on the "Pets" link under the Personal tab in the Navigation Bar. From there you can choose what species of pet you would like, pick what gender you want them to be and give them a name.
If you would like to simply adopt a pet from the pound, head over to Veta Lake to get one. They will cost only a small amount of sP but can come in whatever color the previous owner had left them.