First of all, cats and dogs are NOT natural enemies. They can form exceptionally strong bonds, especially when raised together from young age. When adopting a cat, ask if it has experience with dogs, and as long as the dog has good experiences with cats it will be easy.
Dogs generally will accept the new cat, as long as it has had another cat in the house, with just a sniff to the rear end. This is where we as animal hosts can help, Cats prefer sniffing noses, but also tend to swipe and scratch at over excited dogs. We can hold the cat’s backside to allow the dog to sniff, then gently hold the chest and front paws of the cat, as we turn it to face the dog and sniff the nose. Leashing the dog is also preferred if they are too excited.
There is very little chance of cats and dogs without positive experiences with the other to become friends. The cat will be defensive and hide, while the dog will chase and get scratched on the nose. Over time and with lots of patience and training the dog and cat may be able to reside in the same house. The cat would need its own space, where the dog couldn’t get to it, so it would feel safe.
Ideally, a kitten or young cat with no experience with a dog, and a dog that has never encountered a cat can often times become friends. This introduction will require some separation time, so both feel safe. Allow the dog to sniff the cats safe place, when the cat is not around. Then allow the cat to sniff the dogs area, when the dog is not around. Be sure to supervise the first encounter of the two, the cat may swipe at the dogs nose, allow them to see and sniff each other. They may not the first time, but short supervised visits over a few days should allow them a safe way to get to know each other.