What are the steps necessary to bring a dog from the US to the UK and then to Italy?

I’m trying to research this now and it’s frustrating and convoluted. Just wondering if anyone here has any experience or knows what’s necessary, so I can double-check my findings. We don’t want the new puppy to get stuck at the border somewhere.

Any links or the like would be great, as well. :)

EDIT: To clarify, the family is flying and the ultimate destination is Italy, but there is a stopover in the UK.