I don't have dogs in my household, but my cats have interacted with the dogs at my parents' place when I've brought them over for a visit (my childhood pomeranian still resides with my parents, along with two dachshunds).
It really depends on the dog. I strongly recommend any introduction be done on a leash at first, then you can assess how the dog will react to the cats without putting the cats in harm's way. If you plan on adopting from a shelter, you'll probably want to see if you can find a dog that has been homed or fostered with cats, or is listed as "cat-friendly."
My dog-cat introductions went differently for the two different breeds. My pomeranian (who has a history of marked indifference towards other animals) sniffed each cat a few times and then promptly decided they were of no interest to him, and ignored them. The cats hissed and ran away at first, and Apollo growled for a while, but being ignored got them interested, and before long, Apollo was following Sambu around, trying to bat at his tail like it was a cat toy, and Athena got comfortable enough around him to just go about her day.
The dachshunds were a completely different story, and the primary reason I recommend introducing on a leash at first. Being an instinctual hunting breed, they unfortunately saw the cats as prey, and each tried to lunge at them. They kept whining and straining to get at the cats, so I never felt comfortable letting them off the leash around the cats and keep them strictly separated from the dachshunds when we're over at my parents' place.
So it can go either way honestly. A dog that is comfortable with cats, or a dog not prone to chasing other animals, or possibly a puppy young enough to be safely trained around cats, would probably be fine. A dog with a strong hunting instinct that has shown a habit for chasing, and/or killing small furry animals may be potentially problematic.