My dog had a bit of an issue when the veterinarian's assistant tried to stick a thermometer up his rear-end. He kept turning around to face the person and keep his rear-end safely out of reach. Eventually after several repetitions he sort of made a snapping motion at her and she had to skip taking his temperature.
The funny thing is this didn't extend to anything else. He loved being around the vet and vet assistant and eagerly lobbied to be petted and get treats. He didn't even squirm or make any objection to the vaccination shots. He was just
not going to let anyone stick anything up his butt. I can't say I blame him, really.

As soon as the thermometer was put away, he returned to his usual friendly self.
I felt a little bit of mental aggression toward the vet when I saw the bill, but that's another story.
Anyhow, I wouldn't necessarily say that you need to be worried about your dog around kids just because he is a little aggressive at the veterinarians office. At your home, I'd imagine he feels like he's in a safe environment and that he knows your children are part of his family/pack and feels comfortable with them, and possibly even protective of them. That's a total different scenario from being in a strange place with an intimidating adult stranger trying to get in your face and probe various things.
What I'd say is just keep an eye on your dog and your kids when they interact. Make sure the kids know not to pull the dog's tail or try to ride him like a horse or that kind of thing. Remind them that dogs are uncomfortable being hugged (A dominance gesture in the dog world), they like being petted instead. Make sure they respect the dog and don't throw objects at him or do anything like that that might make him feel unsafe. Maybe don't leave them alone together until a bit of time has passed and you know they'll get along.
Most golden retrievers I know absolutely love children, and children love them. Children that grow up with goldens think of them as their best friend a lot of times. They all kind of run around and play together and tire each other out and have a grand old time. The enthusiasm of goldens as a breed, and their playful natures matches up well with the temperament of a lot of young humans. And children who grow up with goldens often wind up getting their own when they're adults, not willing to settle for any "lesser" breed of dog.
I think you made a good choice for your children getting a golden. It'll create memories and experiences for them that they'll really cherish.
Just remember to use common sense and remember that any pet is still an animal, not a wild animal, but an animal, so it's worth just an eye on things and making sure there aren't any misunderstandings between him and the kids. There likely won't be any problems, but just like you'd watch the kids with a new human playmate or screen a new babysitter, it does make sense to keep an eye on things until you're all totally comfortable with the situation and the dog has become a full fledged member of the family.
