Internships and residencies are always after vet school. Undergrad>Vet School>Internship (usually 1 year, maybe more depending on the specialty)>Residency (3-4 years, maybe more if getting a PhD).
I'm a clinical pathologist and clin path is a little different anatomic path (which lilylilac described well)
In clin path, a rotating internship or some practice experience is pretty highly preferred over joining a program straight out of school, though people straight out of school can land positions (but if everything else equal, most places pick the person who practiced). The internship is usually the same one that the other clinical specialties do where you rotate through ER, internal medicine, surgery, etc. There are two or three schools that are starting to offer pathology internships...as far as I know these programs are split between time on anatomic and clinical pathology and you are often on call to run labwork overnight. They aren't required by any means and kinda let people make sure path is the right choice for them before committing or help them select anatomic vs clinical.
Clin path also places less stress on needing a PhD...more and more faculty positions are becoming clinical track (unlike anatomic academia where a research program/PhD is almost required).
You certainly don't have to stay at the same institution you went to vet school at, and most people don't. Most academic institutions have residency programs. I don't know of many/any private practice pathology residencies. Sometimes a private laboratory will contract with a school where they'll fund the resident's training in exchange for a 3-5 yr committment, but they're still located in towns with vet schools.
From reading your post history, you're still trying to get into vet school. While it's good to look ahead, don't get the cart too far ahead of the horse. Do as well as you can in vet school and worry about residencies once you hit 3rd/4th year. So many people change their minds about what they want to do during vet school.
