Something to feed your pet in will be needed, of course. These don't have to be made just for dogs or cats, but it is a good idea to designate certain dishes for them.
Bowls made for pets do have some advantages. Many have rounded bottoms, and often these are the easiest for pets to eat from, especially if they are eating canned dog or cat food.
Some pet dishes have rubber feet or rings around them to make them "skid-proof", which is also a bonus. Bowls with round bottoms can't be tipped over by over-excited puppies.
When choosing pet bowls, stainless steel or ceramic are preferable to plastic. Plastic can leach toxic chemicals as they age. They are also harder to sanitize, as bacteria can invade the plastic as it wears. Plastic also scratches easily and cracks, and soft plastic can be chewed up.
A collar and leash is a necessity for dogs, and an option for cats. Cats can also be taught to walk on a leash if you wish to do so. If you plan to let your cat go outside (not recommended), than a collar is a necessity, not an option.
Identification of some kind should be firmly attached to collars of both dogs and cats. Be sure to include at least your phone number and the name of the pet.
A place for the dog or cat to sleep is nice. This may mean a place on your bed, or a bed made especially for them. Older dogs and cats might benefit from orthopedic bedding. This is usually made from "egg-crate" foam. You can buy a premium bed, or make your own from a piece of the foam and an old sheet or blanket.
Cats will often ignore any bed you buy for them and choose their own spot - usually your favorite chair or the middle of your bed. However, there are a number of cat beds available that made for the way cats like to sleep. They provide small, snuggly spaces, or are tent-like, offering a nice place to hide.
Most dogs will benefit from having some toys, and for puppies they are a must. Interactive pet toys allow you to spend "quality time" with your dog or cat while providing the animal with some exercise. These would include balls for dogs and wand-type toys for cats.
Chew toys are essential for puppies. Their natural instinct is to chew, and puppies go through teething just like babies. Here is one area where buying new is preferred. It is not a god idea to give the puppy an old shoe, for example, to chew on, as that would imply that all shoes are puppy chew toys.
Squeaky toys can drive owners crazy, but they are good toys to leave for your dog when you go out. Many dogs enjoy throwing these toys up in the air and then pouncing on them. Just be sure your dog cannot easily chew the toy up and swallow the squeaker. Always supervise your dog when giving them any kind of new toy, and make sure that it is appropriate for that dog.
If you are planning on getting a new dog or cat, it is wise to have all the necessary pet supplies on hand before you bring the animal home.