You could train as a pet groomer/bather. Also check with some of the colleges/universities to see if they have correspondence courses or internet-based courses (make sure they're legit programs, though).
You could also volunteer at first. Since you said your mother's in a nursing home, check out the Delta Society's website about animal-assisted therapy. You could use your own pet or see if a shelter would have an acceptable animal you could train for it. My cat and I visit a nursing home twice a month to cheer up the residents.