My wife thinks puppy first, I’m thinking the kittens should be first.
This is more about them, the animals, than us. We are going to love them to bits whatever we decide.
We have finally found two kittens (boy and girl) that we will be able to have mid-September. We have also found a puppy (boy) that we can have the beginning of August or we can wait for the next litter (boy or girl) which means December pick up.
Generally I think it is better to introduce a kitten to a dog, as a kitten it is not afraid... rather than a puppy to a cat, where it's more likely the cat would be fearful. However if it's just a couple of months difference, it might not matter.
I had cats in the beginning and then added a puppy. I think it worked pretty well because the cats were established (is this the correct word?) here and when the puppy came, they showed who is the boss. It only lasted a few days. By the end of the week all of them were sleeping together and never had a fight.
I'd previously had a dog for years, then introduced a kitten.... she still grew up to be the boss even though she arrived second
From a purely practical standpoint, if you and/or your wife are currently working from home, perhaps due to COVID, and you would have more time and flexibility in August to train the pup, that might be something to consider.
In my experience, bringing a puppy into the home is typically more chaotic than bringing in a kitten, however I have only ever had one kitten at a time. Sounds like your household is going to have a lively future!
If puppy/kitten grow up together then they often have a stronger bond, except the cat is always the boss! Kitty refused to play the "chase the cat" game with our old boy but just sat there and stared at him when he wanted to chase her. She would also plonk herself in the middle of his bed and not move, forcing him to sleep on the edge/on the floor.
Another thing to consider is the size of the dog and how much the dog will have grown if you introduce a kitten a couple of months after.