Tl;dr - Dogs decide a lot of things solely by how it smells. Let the animals get used to the smells of each others, and keep the dogs calm no matter what the cat does.
The long version:
I have brought a cat, and then some more cats, to our home where we already had two dogs for several years as our only pets. Every cat was brought in the same way.
First the cat was brought in our home in its container letting the dogs take a quick sniff of the container and its inhabitant through the narrow slots, then I took the container and cat in one room and closed the door. Our dogs were quite cool during the brief moment, possibly because the container was in my hands, and our dogs tend to keep their calm better with me than with my wife.
Once the cat is in that one room, it was let free by opening the container door. Some cats take longer to come out, some sprint out pretty soon, point is that the cat comes out voluntarily. That's it for a couple of days. The cat will get to know you, and feel safe in the room. It will find food, water, litterbox and normally it will feel at ease soon enough. Depending on the age of the cat, and its personality, this phase could take longer than two days, but two days has been enough for us so far.
Next we have taken the cat into livingroom in my wife's lap, in such a moment when the cat has recently eaten and is calm, sleepy even. Gone to sit in a sofa with the cat in her lap, so that our dogs can come close and sniff the cat. (Our dogs are not allowed on the sofa, they stay on the floor.) My wife holds the cat so that it can't see the dogs, but dogs see the cat, or at least its rear end. If dogs become excited, I'll take them into another room and leave them there. Goal here is to let them get the smell of the cat, and to see it protected by a human member of our family. We let the dogs sniff the cat as much as they like, just as long as they do it nicely and stay calm.
Repeat step 3. a few times, a couple of times a day or even more. This is to see how the dogs react, and as soon as they are cool with seeing the cat then my wife has started letting the cat see the dogs too, and perhaps even let it move with some freedom on her and on the back-rest of the sofa. Kittens are nice in this phase, adult cats might be more eager to seek a hiding place. We have returned the cat into its room once it starts to turn playful or my wife gets tired to hold the sleepy cat.
While the cat is in livingroom with my wife, I let the dogs in the room the cat has for itself. I only remove the food first, then the dogs may sniff around as much as they wish, drink the cat's water if they like, just as long as they want and stay calm. If they get excited, it's the penalty room again for them (never a long penalty, just for the moment while the cat is still in livingroom with my wife).
The better the dogs behave, the faster the cat is let more freedom to explore the livingroom. With our cats and dogs this phase has come in less than five days, but I'd guess this is because we've been bringing kittens in house. It may take a lot longer with adult cats, but I've never done that (yet).
Later, when the cat is fully accustomed to its new home, we have still needed to keep an eye when our dogs are eager to take part, or to shepherd, the cat when it is playing. We discourage the dogs to take part in the play. At the moment we have five cats and sometimes they play quite roughly by themselves. Our dogs tend to "go looking" when a cat is complaining over too rough play. We just tell the dogs to let the cats be, and that's it. I can see the dogs would still like to go and see what's bothering the cats. Maybe they do it while we are away, however I can tell the cats don't just stay up on their platforms when we are away, there's always clear signs of cats having been playing around the house when we come home :)
This has actually been the easy way, when our cats have come in as kittens and our dogs have been more than three years old. Younger dogs can be very different, for they are still "teenagers" and have lots of extra energy, and an older cat can be a handful as well.