When your Sim clicks on a horse you'll see a Care button. The stable already has hay in it for eating. The other needs are pretty self explanatory - have a trough of water and a salt lick near the horse's stable.
That does not necessarily mean you always have to ride them, but you need to be on your toes and monitor their needs and give them some self direction to gallop on their own. Horses do need a lot of exercise, that's for sure.
Most of a horse's 6 needs are the same as a Sims' needs (Hunger, Bladder, Energy, and Social), but instead of Hygiene and Fun, horses have Exercise and Thirst. The other options to getting a horse are 1) using a phone, computer or newspaper and adopting one for free, 2) befriending a wild horse (you had better be patient for this option), or 3) simply going to the Equestrian Center and buying one. There's no shortage of customization options available in Create-a-Pet. The game also will throw a little extra money at you for each horse you create since you'll need to buy supplies for its care. The advantage here is you get to choose some traits that will make things a little easier and more efficient for you, like Agile (for the jumping skill), Fast (for the racing skill) and Genius (for learning skills a little quicker). If you're starting a new game, then obviously the easiest way to obtain a horse is creating one at the same time you're creating your Sim(s). In this overview guide, I'll lead you through the basics of horsecare, learning the riding skill, and finally, I'll touch on the new career as well as the competitions. Dogs and cats (and the other minor animals) are a big addition too, but the whole new horseman career, the new riding skill, and all of the competitions you can enter at the Equestrian Center really puts horses in their own category. Of the entire scope of new content in Sims 3 Pets, horse related gameplay is unquestionably the deepest and most varied. The Sims 3 Pets: Horses Guide Basic Horsecare, the Riding Skill, a New Career and More! by Tom "Metro" Schlueter The Sims 3 Pets Guide