I successfully intro'd my dog & cat. The dog has an EXTREMELY high prey drive and the cat is a bit anxious and was definitely not excited to have the dog around. It took a long time but they've been great now for about 3 years. The key is not to rush them and never force the cat to be around the dog.
*Warning..this is a long post for what could be a very tedious/long/stressful process. But, it worked. I'm not a trainer but I read a lot and this is the solution that I came up with that worked for my kiddos.*
Step 1: Separation
First, separate them by a baby gate. Ideally, it's a gate they can see each other through easily & is screwed into the door frame (for their safety and this will ensure the gate never gets knocked down). The area near the gate has to be all about positive behaviour. No form of bad behaviour is acceptable near the gate and (in the case of an anxious, not aggressive cat) if there is bad behaviour by either party, the dog is the one that needs to learn to move away. If the cat is aggressive, your job will be much harder but I still believe it's possible..more on that later. For now, we'll assume the dog is the one with the bad manners
When dog goes out for walks, cat can be allowed to roam freely around the house, as long as you can safely get cat back into her gated space when dog comes home.
Step 2: Building Positive Associations for Cat
Now that they are separated in their own safe spaces, I would put a blanket or toy of the dog's in the cat area so that she can start smelling and familiarizing herself with dog. You can even give her treats near the dog's smell. She might not go for it right away..give her time. The goal is that she learns that the smell of dog can be a positive thing.
In the cat's gated area, you should also move her food up to a secure dresser or table top, etc. This will start teaching the cat to climb up for food so that, in the future when there is no gate, cat knows she has a safe place to eat without worrying about dog. It also provides her a nice bit of extra exercise
Step 3: Friendly Gate Visits
I would then start visitation sessions near the gate by sitting on the floor on the dog side of the gate and calling the cat over.
Cat might not approach the first few times but with any luck and possible some extra motivation (treats), she'll start coming by the gate.
As the cat approached, I would tell the dog to "sit" and then "lay down". When the dog shows good, calm behaviour near the gate, reward. Same goes for the cat. If the cat comes near the gate and shows positive behaviour, reward (with treats or verbal commands). If the dog begins to get too excited when the cat is near the gate, move the dog to another area and repeat "leave it" command. Try again later.
If the dog is getting too amped up even just being in front of the gate, I would leash dog in the house somewhere securely away from the gate until she was calm. As soon as she was calm, I would unleash dog and repeat as necessary. Our dog was very receptive to training this way so this lasted not even a day before she got the hint. Some people will argue that this isn't good training because it's essentially punishing the dog for bad behaviour rather than just ignoring it but in my view, hunting the cat is completely unacceptable and ignoring this type of behaviour wasn't a risk I was willing to take.
Step 4: Preparing Dog for Open Cat Time
Next, I wanted to establish a command with the dog that redirects its attention & teaches self control in an even more structured way to prepare for future interactions when there is no gate.
I did this by teaching the dog to "go to your place". I would repeat "go to your place" and then say "good place" when the dog got there (a dog bed), reinforcing with treats and rewarding her "stay".
Eventually, you can use "leave it" and, if needed, "go to your place" if the dog gets too excited near the gate.
This gate visitation phase & "go to your place" training went on for a couple weeks. Yes, weeks..probably not what you want to hear but slow and effective is better, especially with cats.
Step 5: Open Gate Cat Time
Once the dog showed consistently good behaviour and self control around the cat with the gate, we started leashed, open cat times. Open cat time means, the gate is open and the cat is free to roam and the dog is leashed to you. The cat gets to wander around the home and the dog has to practice self control. When cat is out, dog remains long leashed to you and is asked to "lay down". If the dog gets excited, you can say "go to your place" and bring the dog to it's place. If still too excited, move the cat back to cat's gated area, continue with Step 1 and try Step 2 another time.
Step 6: Testing Open Dog & Cat Time
Once the dog shows self control while leashed to you and is able to follow commands while the cat is roaming around freely, we began to test open cat & dog time where the gate is open, cat is free to roam and dog is off leash.
During this time, it's very important to stay near the dog and ensure you either have a collar or untethered leash on the dog so that you can easily gain control if needed. Just because dog learned to behave on a leash, does not mean dog will behave off leash. This is very important to remember.
I found the most success by teaching the dog that she had to stay in "her place" while the cat is roaming about. If the dog couldn't follow this command while the cat is out and about, I would separate them, go back to the previous step and try again later.
Eventually, dog learned that she had to stay in her place when the cat was around.
Then, I started leaving more space between dog and I to see that she could control her impulses while I wasn't right next to her and eventually we moved to the final phase.
Step 7: Take the gate down!
Finally, the gate was staying open longer and longer and eventually, we took it down!
Just remember, it might take more or less time depending on their unique personalities but it can be done! Just be patient (really, really patient) and be consistent with your training.
If you have any questions, ask away! Otherwise, good luck!