Until you boost your Sim's riding skill/ the horse's racing skill to level 1, that horse is going to be very slow. It won't take quite as long to ge to the grocery store after that. In the meantime, taking your horse for a ride on every errand will help you build skills. But training on the training posts at the nearby Equestrian Center will be faster. And you won't need to bother buying training posts or wall jumps for your home lot.
In a similar situation, I sold some of the houses furnishings to pay the bills until she started winning races. Also, adopting free horses from the shelter, training them, and selling them is a pretty easy way to make money with horses. And you can feed and water them for free from hay bales and water troughs at the equestrian center, and hitching posts and ponds around town.