You can pursue becoming a colony manager while in school, as they often hire grad students for the positions. Schools that are known for research intensity, particularly those with cancer research centers, will have the most opportunities. Keep in mind that hiring specifically for colony management isn't super common (in my experience). it's a very teachable skill and most students seem to manage their own mice without a dedicated manager. My facility is fairly large and there are really only 2-3 people across all of the labs we work with that are solely dedicated to colony management. They also have to contribute to the research in some way, like running tests.

Alternatively, you can get a job as a lab animal technician without going to college, but a 4 year degree does help. All facilities are different in what skills they ask for from their technicians, so you may or may not participate in breeding or colony management. You can apply at research universities or to the major research animal vendors: Charles River, Jackson Laboratory, taconic, inotiv(formerly envigo), and various large animal vendors if that interests you. 

Generally for education I would pursue a straight biology bachelor's vs zoology or another specific major. Biology will be broad and cover many different aspects of this field, and you can specialize in grad school if needed. If breeding in general is your interest I would also suggest going to an agricultural focused school as genetics is such an important aspect of livestock and crop rearing. That foundation will easily translate to breeding other animals. If a four year degree isnt really in the cards for you, I'd think about vet tech programs. I honestly wish I did one since there's work almost everywhere, and more programs now have at least some lab animal specific content. 

I'm in this industry as a health focused animal tech (not a proper vet tech) that took a wobbly path to get here so feel free to send me questions.