Dog grooming and horse grooming are very different things. I think you will be hard pressed to find someone who will pay you to simply groom their horse.
Clipping yes - but you need to be very good at clipping with no lines, and have a great deal of experience in dealing with nervous and difficult horses.
A 'groom' in horses, is someone who works for a rider/trainer, and takes care of the day to day needs of the horses - a small part of that will be brushing them and keeping them looking tidy. The rest will be mucking out, cleaning tack, cleaning boots, washing gear, moving feed, handling often difficult horses, often trailering horses to and from competitions/appointments.... and long, hard work days. Most grooms start work at around 6-7am, and work through unti 6-7pm in the evening, 6 days/week, for less than what you'd be paid to work on a checkout.