You know your resident pet well, of course, so it may be easier to watch them for signs of upset, but keep a close eye on your new pet, too. Lowered tails, pinned back ears, crying, shaking, etc. indicate discomfort, anxiety, or fear.
Best Friends has some helpful insight on what body language cues to take note of as well, sharing that "If your dog has a strong prey drive (the inclination to seek out, chase, and potentially capture animals seen as prey), they might become very focused on the cat. The dog will stiffen, stare, and possibly bark or whine.
If you see these signs, do not let your dog near the cat. Ideally, the dog's body language will be loose and relaxed around the cat. It's OK if your dog pays attention to the cat, but you don't want to see a dog fixated on a cat."
Are you planning on having an indoor-outdoor cat? Pay attention to their behavior when your pets share space outside, too! Best Friends makes a strong point, "Just because your dog is comfortable with the cat inside the house, that doesn't mean the dog will exhibit that same behavior outdoors. They might fixate on the cat and start stalking or chasing the cat when they are outside together. So be aware of your dog's body language around the cat in each new situation until you know how they're going to respond."
Respecting your pets' boundaries and paying attention to the cues they are communicating is essential to keeping them happy and safe. Forcing a situation on them that they're not comfortable with is a recipe for disaster. Instead, follow their lead when necessary, and separate your pets when it's in their best interest.