My wife and I want to eventually have a cat, a dog, and a bird. I just got to thinking, which order would be best to get them so that they all get along?
If you get the cat/dog as kittens/puppies, they’ll usually get along just fine with the other. Or the other option would be to adopt a grown cat/dog that was exposed to the other and is fine with it.
Bird, on the other hand - I don’t think there’s any way to keep the “Kill! Kill!” instinct out of a cat or dog when it comes to birds. I’ve had cats & dogs with birds, and pretty much if the bird was out of the cage, the dog/cat hunted it.
Or, in the case of cats, they spent most of their days sitting under the bird cage, trying to figure out how to get the bird. I have one really cute pictures of a 2-3 month old kitten clinging to the side of the bird cage, after taking a flying leap from the couch that was a good 5 feet away.
The hunting instinct is pretty difficult, if not impossible, to get rid of.
That’s what we’re worried about. The only thing I can think might mitigate that is if the bird is bigger (both older and physically larger) than the cat/dog when it is brought home.
Thing is, my wife doesn’t want parrot, but something like a little parakeet (we’ve decided on a compromise:cockatiel)
I’ve never had birds, but with cats and dogs, I have had much better luck getting the dog first and introducing a kitten to it. I’ve been having a heck of a time with my older cats and my new puppy. They just don’t care for my pup’s enthusiasm.
I’ll tell you how to get a dog that won’t try to kill your bird: get my mom’s bird.
She has Paulie, a parrotlet (hers looks like the blue one). That’s her size fully grown.
Ziggy, the cocker spaniel we’ve had 9 years, is terrified of this bird. He refuses to go near the bird and if you take Paulie out, he will leave the room. We figured that was just Ziggy. Then they adopted Buddy (another cocker) who is even more terrified of the bird than Ziggy. He literally will run away to the bedroom, shaking in fear.
This is a 3" bird, for crying out loud!
Now, I have no clue about a cat. 
Since you can only safely transfer one animal at a time to the living room, take the cat and leave the dog and the bird in the garage. They’re basically safe together since the bird can fly and the dog can’t climb the way a cat can. Then leave the cat there and go back into the garage. Call the dog to you and leash him and take him into the living room. Command him to lie down. Obtain the cat. Leave the dog and go back into the garage. With you to protect the bird, the cat will behave himself for the moment. Go get the bird. Leave the cat in the garage and bring the bird into the living room. The bird will be safe with the dog. Now you can go back to the garage and get the cat.
…What?
Damn you! You stole my bit! 
I got a dog - when the dog was just a year old, we got a kitten. that worked perfectly because the dog was trained but still “puppy” enough to play with kitty.
They are best friends and curl up together to snooze.
If you get one of those dog that doesn’t give a hoot, I don’t know what you call it, but you know how you have an alpha dog and a beta dog, you can get one that is “Z” on the alphabet. 
A dog will cope with a cat much better than a cat will cope with a dog, though as another poster said, I think it depends on how they were originally raised. I had a cat and a chinchlla and the cat used to sit for hours and just stare at the chinch. I never trusted the cat alone with him.
So you have to look at the individual dog. Probably a pound or adoption shelter could tell you what kind of dog or cat you’re getting and how likely it is to cope with your lifestyle
I’ve had several combos of cats and dogs growing up…generally speaking, the cats were the “ruler” of the dogs, regardless of when the cat vs. the dog joined the household. The dogs seemed to view the cats as potential playmates, and generally the cats were too bored/annoyed by the dogs to play along.
Lots of “cat sitting on sofa/dog walks over to say hello/cat either ignores dog or takes a swipe at dog” situations.
I’ve never had a kitten and a puppy at the same time; I would imagine their relationship would probably be friendlier. That said, the situation never escalated to openly hostile or worrisome. Most dogs IME are either afraid of the household cats or want to be friends with them; this idea that “dogs chase cats” only applies to strange cats, at least in my experience.
As for the birds…I used to keep finches as a kid, and the cats were always fascinated by them, and would actually sleep on their cage if allowed. (Zebra finches are kept in cages with bars too close together for a cat to wedge a paw inside so it wasn’t a big deal.)
I can’t imagine there would be many situations where a cat WOULDN’T want to stalk a bird; as long as the bird is kept out of reach I don’t see a potential problem.
Beat me to it.
Get the bird first. It shares the least in common to the other two and will be the odd man out by default - it needs to have already been part of the new environment that the other two are beginning to accept as their new home.
Get the dog next. It could be immediately after the bird, but only if a bird in general isn’t new to you. The dog needs to be there before the cat for similar reasons to the bird being first: when the cat arrives, the bird and dog are just parts of it’s new home.
Cat last. Introducing the cat last probably won’t make it hate the bird or dog any less, or reduce it’s attempts to kill them both, as cats are evil. But it will give the other two a bit of a head-start finding hiding places around the home.