Most of the dogs I've owned were bought as puppies except one who was a stray we found. When I was a kid we adopted a cat from a shelter. I don't really remember the process, but looking at their site in present day, it looks like you just need proof of identity and address.
I have a dog and want to get and elder Italian Greyhound next. I want to be more specific and careful especially because my current dog is old and small himself, so I'm not prepared to take in a dog from the pound just because of the unpredictability. But for breed specific adoption, those involve rescues and from what I've seen it's very involved. References, home visits, places saying it's cool if you rent but you must have a backyard. It's disheartening to think I could do things right and they'll still say no. I've just seen so many horror stories online about weeks/months long adoption processes that ended with no dog.