In BitLife, you will need to get a dog from one of the options under the Pets activity tab.
Under pets, you may get a dog from either the pet store or a specific dog breeder that sells specific breeds of dogs.
As there is a far greater selection of dogs and a wide range of statistics per dog, going with dog breeders might be the best option for you.
If you want a dog from a dog breeder or a pet store, please click on either of them and then look for a dog. For the pet store, you'll need to look the list for one.
Just tap on a few to see if you can purchase them. For dog breeders, you will first have to select a breed you want and then tap on it.
You will need to select a dog from a list that falls under a breed of dog after tapping on a breed of dog from the list.
Tapping on any of the listed dogs will allow you to purchase them in the same manner as pet store dogs.
On Pro Game Guides, check out How to Bring Pets to the Vet in Bitlife or How Many Pets Can You Own in BitLife?