The equestrian centers have food/water stations to feed a horse, so you can care for it. But unless you have a park or your sims lot with racing/jumping training items, it may be hard for your horse to gain skill to be competitive in high end races.
Though if your sim is up for lots and lots of long runs back and forth across the town, you can get a horse's racing skill up that way.
Another way to get money with a horse is to have a mare and breed her. Once you've trained the baby, you can get several thouand dollars for it.