In the console version of this game, you cannot get babies, as there are no children, but you can breed your pets.
First, you need a male and a female pet, and they have to be either both cat or both dog. Infact, you don't even have to own both of them! Yep, you can just get one, and hope a local cat or dog of the opposite gender pay you a visit.
To breed your pets, you need a special kennel type thing. You can get one from the object catalogue or you can wait and buy one from the pet emporeum. Then, place it in your lot, and have your sim tell your pet to 'Try for offspring with...' and then pick which pet you want to try their luck with. If both animals are in a good mood, they will go into the special breeding kennel thing, and love hearts will float out the top for a while. When they come out, a puppy or kitten will arrive in a hot air balloon. You can name your new pet now, or wait until they are about to grow up!
Puppies and kittens grow up in (approximately) one sim day. Unlike in the PC expansion pack version, the offspring stay in their baskets when they are young. When they grow up, you can keep them as part of the family, or sell them for pet points in town.