Food items can be bought from the Food Shop, obtained from events, or found in Prize Boxes and Daycare Gift Boxes. They can, of course, be fed to horses of any stage, and will grant a number of levels specific to the food item that was used. Food can give anywhere from 1 to 149 levels at a time. Unlike on some other sites, there is no penalty for not feeding your horses, but feeding them can help them reach adulthood faster.
Below is a list of food and how many levels each gives.