I worked full time at the local no-kill shelter here for five years. I did intake of new animals, cleaning, medicating, driving to vet's office with them in my own car, and handled adoptions. I just made sure to leave my smock at work where it was washed daily, and washed my own clothing right away. I also changed my shoes before I went home. I had a dog and a cat when I started working there and I never brought home any diseases .... I did however bring home a puppy, many orphan kittens, and a few dogs that I fostered!
I think that it's great that you want to volunteer