Important thing to know is that it really doesn't matter what you major in--you could even major in art if you wanted to, and you would still be able to apply to vet school as long as you make sure to get all the necessary prerequisites out of the way too (which vary from school to school, but of course typically include a lot of bio and chemistry courses). You would have to look up the specific prerequisites for whichever school(s) you are planning on applying to.
Also, if you are interested in vet med as an option, you'll definitely want to look into shadowing with a veterinarian in order to start getting an idea of what the profession entails! Animal shelter volunteering is good, but unless you were directly volunteering with a shelter VET it doesn't really tell you much about veterinary medicine, just animal care. As many people here will tell you, it's important to take the decision to pursue vet med into deep consideration, because vets face a very high debt-to-income ratio compared to human medicine, so if you would be equally happy doing either than it's financially better to stick to med. You could definitely continue doing shelter/animal volunteer work whilst pursuing med school, too.
As for the main question, I have a degree in animal science. I chose it because the coursework was more interesting to me than a regular Biology degree, and still involved many of the same biology classes. While I can't speak for everyone, I have been able to apply for a wide range of animal-related jobs and internships (since they typically require a degree in Bio, Zoology, or any similar animal/life-science related field). I actually just got accepted to one internship at an aquarium which I will be undertaking this summer. You'd also be eligible for careers in various fields of agriculture. Here's a nice general overview I found of some career examples that would be available with this degree:
https://asas.org/membership-services/member-information/career-information/animal-science-careers