Ideally it’s best to introduce a dog to cat when both are little. But in most cases one is older than the other. If the cat is older and you just brought home a cute little puppy. It is best to keep the puppy on leash, because any sudden movements will get the puppy excited and want to play, which the cat may take offense to the puppy and attack it.
Make sure for a couple of weeks that you never leave the two alone, always monitor them and praise both of them for good behavior and reprimand for bad behavior. Make sure that you provide a good amount of treats to both of them for praise.
Make sure that everyone in the room is calm. The surrounding area needs to be calm and peaceful for the introduction. That is including you, if you are scared or fearful both animals will be able to feel it and more likely it will be for them to attack one another. If you need to have a cocktail or go to yoga before hand then do so, it will relieve much stress and make the introduction a more pleasant one.
Since you are introducing dogs which most have tons of energy, it is best to make him/her run a bit before the introduction. Running will release a lot of the dog’s energy, which will allow for a better and less energetic experience for the both of them.
You have to remember that their two different animals and they think differently and will want their space. Make sure you set up private spaces for each animal. A space where each feels comfortable and safe from one another and neither of each can get access to the other’s space.
You need to be consistent. It will take you many introductions before you get the results you want. Don’t give up and don’t rush either of them. It is best to be patient in this process; you need to make sure you trust both the cat and dog. There will come a time either they will love each other or learn to live with each other. In all it’s a long process.