I studied Genetics at Uni! Majored in it actually as well as Biochemistry.
You best bet for that type of work would be to study Genetics as well as perhaps Zoology or even Evolutionary Biology. You could then go on to do research which can be a little tricky to find work in a field as specific as Equine Genetics but possible I am sure.
Here in the States the salary for working as a Research Assistant starts at around $30,000 I think.
Actually, to give yourself a better chance you would also have to consider going to graduate school to do a Masters or PhD in Genetics as this will give you a much better chance of finding work after graduation. The way most of these programs work is that you spend 1-3 years doing additional classes then complete a research project and write a thesis on your topic. So if you are accepted into graduate school and can find a current researcher there that specialises in something equine, you would find out if they would be willing to have you as a student.
But first things first - Undergraduate degree with a Genetics major I say!
All horses deserve, at least once in their lives, to be loved by a little girl.