As in most Sims games, you are able to breed your pets. Puppies and kittens can be sold just like your other pets by clicking on the sim you want to sell them too and selecting Sales... and then the name of your pet. If your sim is particularly generous, they can also give the pet as a gift by choosing the action Give Pet on another sim. This only works for sims from other households.
Breeding your pets
First off, you need either a male and female dog or a male and female cat. As pets can be altered in this game it doesn't matter on the breed; however they can only breed with pets that are the same species. Next you need a pet house. These range from the dirty wooden kennels to the grand cushy-pillowed pet houses. Finally, you need to build the relationship between the two pets. There is a way to do this quite easily; wait until one pet has learnt the Playful Behaviour, and then keep commanding it to play with the other pet.
Once your pets have become good friends, it's time to use the power of pet influence. Simply by selecting Try for Puppy with... or Try for kitten with..., your pets will retreat to their pet house to mate. If you hear the chiming music when they come out, you know they were successful, and very soon the mother will give birth.
Pets gestation periods are about 2 days; less than Sims which are three days. The litter can range from one to four puppies or kittens, but you cannot have more than six pets in your household. If you already have that many, your pets will not be able to breed.