It depends on the personalities of each animal involved.  I've had two types
of dog-cat experiences, and I expect neither of them is necessarily typical.

When I was in my 30's, I had never known a cat up close and personal.  We
always had a dog when I was growing up, and my husband and I had a dog
during much of our marriage to that point.

At the time, we had a male beagle, about 5 years old.  He was ingenious at
getting out of the yard.  One evening he had gotten out, and my husband Jim
had gone out in front to call him.  Sometimes this worked.  That particular
evening, the dog, Max, didn't come, but a scrawny, dirty red kitten ran up
to my husband as though he were the one being called.  Jim brought the
kitten in to show me, and I petted it, but we both agreed we didn't need a
kitten, especially since we had a dog, so he put it out before we went to
bed.  In the morning when Jim left for work, he stepped out the front door,
and the kitten, who had been lying on the porch, ran over to greet him.  We
had ourselves a cat, which we named Tawny.

When Tawny and Max met, I was sitting in the living room, with Tawny on my
lap.  Tawny was purring.  Max entered the far end of the room, saw Tawny,
and walked slowly toward us.  Tawny saw Mac, and the closer he got, the
louder Tawny purred.  Max walked up to me, put his front paws on my knees
and began washing Tawny's ears.  From then on, both animals acted as though
Max were Tawny's mother.  We weren't allowed to touch Max's ears, but Tawny
could play with them.

Tawny grew up to be quite a fighter.  He terrorized all the cats in the
neighborhood, and many of the dogs as well.  Eventually Max went to the RB.
Later we moved to another neighborhood.  By then, Tawny was about 10 years
old, and when we moved, he didn't have a reputation to keep up, so he stayed
closer to home and became much more placid.

Eventually we acquired other cats.  One day a skinny, wobbly kitten came
through our chain link fence into our yard.  He was so small and weak, I
thought he was a very young kitten, but the vet said he was 4 or 5 months
old.  He had been mistreated and was starving.  He apparently had brain
damage as a result of some of the treatment he had received.  He was sweet,
fearless, and far from bright.  We named him Charky.

One day my parents came to visit, and brought their terrier cross, Betsy,
with them.  Betsy had the idea that cats were to be chased.  First she
caught sight of Charky, who was lying by the side of the house.  Betsy ran
toward him, barking.  Charky just lay there and looked at her.  She ran up
to him and shoved her nose against his side.  He just lay there and looked
at her.  She gave up and walked away in disgust.

Later, she saw Tawny, and chased him behind a bush.  A minute later, Betsy
came out from behind the bush, thoroughly cowed, and with a scratch on her
nose.  For the rest of the visit, she didn't see any cats.  ;-)

Joy