PUPPY:
Crate: It is difficult to express how important a crate is. Crate training is a spectacular gift sent from heaven. Puppies like to have their safe home to live in and it also makes traveling easy. Puppies learn not to go to the bathroom in their sanctuary and grow to love their crates.
Nylon Collar and Leash: Puppies are small and fragile. You don't pick them up by their throat, so why would you buy an uncomfortable metal or solid collar to choke them with? Buy a soft, flexible collar. They are cheap and better for the puppy.
Water and Food Bowls: Preferably stainless steel. Indestructible.
Toys: Puppies are famous for chewing on furniture and shoes. Buying a small collection of toys will distract them from such temptations. Recommendations: a kong (time consuming toy), a plush, squeaky toy (leisure toy), a plush soccer or basketball (sport, solid ones might take their teeth out prematurely).
Food: Feed your dog top quality food. Since he or she is a developing puppy, they need to be nourished properly. Puppy portions and food types are different than those of adult dogs.
Vet: Make an appointment with your vet to make sure your puppy is completely healthy.
Medications: Buy flea and tick preventative medicines.
KITTEN:
Just like puppies, kittens need food, vets, and bowls (referenced above).
Scratching Post: While this seems optional to many people, even a small scratching post can save your couch from utter destruction.
Litter: Cats go to the bathroom in a litter box and it is best to train them as soon as possible. Clean the litter box frequently to keep your house and the kitty sanitary.
Toys: Kitty toys are a little different than puppy toys. Kittens like to chase toy mice around and play with jingling bells versus chew on teddy bears.