'Fighting like cats and dogs' is a commonly used phrase to express discord but it doesn't represent what life with both dogs and cats must be.
With a little help, a cat can be introduced to a dog's household ; I know; I recently added two six month old kittens to my three dog home and all is going well. Dogs and cats are totally different pets but for those who love both, here are some tips to create a peaceable kingdom.
Refresh Your Dog's Training
Before bringing home your new cat, refresh your dog's obedience training. A cat will be distracting and exciting; if your dog will listen to you and cooperate when you ask him to then the transition will be easier. If he doesn't cooperate, he could potentially be dangerous to the cat's safety. Make sure your dog will sit, lie down, and stay as well as 'leave it' (ignore distractions). If your dog isn't listening well, consider enrolling in a dog training class or ask a dog trainer for help.
Visit a Cat
If at all possible, introduce your dog to a friendly cat; perhaps a neighbor or a friend has a cat who lives with a dog. Your dog doesn't need to touch the cat and in fact that's not a good idea right now. Instead, just let your dog be in the cat's vicinity on leash and watch, smell, and listen to the cat. What you need to determine is whether or not your dog will be friendly to a cat or look at the cat as a potential snack. If he's too excited, is staring at the cat while quivering, is lunging, barking, or growling; please rethink your desire to get a cat. However, if he's excited but not overly so, and calms himself or allow you to calm him; then this new addition to your household might work.
Choosing a Cat
The cat you choose is an important part of this process. If you're going to adopt an adult cat, talk to the shelter's staff or rescue's volunteers and if at all possible, choose a cat who has been previously socialized to dogs. An adult cat who is afraid of dogs or has never met dogs might never relax enough to be friendly to your dog. However, if you adopt a kitten you can raise the kitten with your dog and although this will take more time, you can nurture a friendship between the two.
Give the Cat a Safe Room
When introducing my two new half-grown kittens to my household, I put a screen door on the spare bedroom and I use that as the cats' safe room. It's important the cats have their own space without the dogs having access. They're fed in that room, and have their cat tree, toys, and beds there. The dogs can sit outside the room and watch the cats and while the cats sit inside the room and watch the dogs. It's a great way for everyone to get to know each other plus it keeps the dogs out of the cat food and litter box.
Opening the Screen Door
I kept the two species separate for about a month. Then, when everyone was used to each other with the screen door in between them, I leashed the dogs and asked them to do a down stay in the living room. Opening the cats' screen door, I invited them out. As they explored I remained near the dogs and praised them for holding their down stay and behaving themselves. After a little while I enticed the cats back to their room with a plate of food, closed the screen door, and then told the dogs how wonderful they were. I repeated this exercise several times over the next two weeks; keeping the sessions short.
When the kittens could come out of their room and all involved (cats and dogs) were calm, then I allowed the cats out of their room without asking the dogs to lie down and stay. Dogs and cats could mingle. However, as this happened, I taught the dogs some rules. Chasing is never allowed and a dog who chased the cats had to immediately do a down stay. If a dog got too excited, he also had to do a down stay. When I couldn't supervise the group, when I left the house, or at night the cats went back to their room. I also continued to feed them in their room so they had motivation to go back there.
Sharing the House
It took me two and a half months to completely introduce the cats and dogs to the point where I'm comfortable with them being in the house together unsupervised. However, my dogs had already lived with another cat and were safe with her. If your dog hasn't had any previous experience with a cat, take as much time as you need. Ideally, you want to create a peaceable kingdom where the dogs and cats can both live together without tension.
Don't be afraid to have high expectations for your dog (or dogs). Make sure your dog understands what his rules are with the cat. He should not chase the cat or play roughly with her. He also needs to cooperate with you when you ask him to do something even if the cat is exciting. If at any time you're worried about their interactions, put the cat back in her room. If you're concerned that your dog might not be safe with the cat, call a dog trainer for help.