Horse care, like most anything else, varied wildly through pretty much every time period.

With that in mind, there were a few universal things.

Whilst on campaign, the cavalryman/trooper/dragoon/person riding the horse would typically be responsible for horse's day to day care. The individual would either know how to care for the horse from their knowledge in non-military life or would learn when placed in a cavalry unit. This kind of care included things like feeding grain, providing water, and ensuring that the horse has been groomed properly.

"Grooming" in a loose sense involves all of the activities necessary for the daily maintenance of the horse. This would be cleaning the dirt and sweat from the horse's body and removing dirt and debris from the horse's feet. This might also entail things like the shortening of the main and tail, as well as washing the saddle cloths and cleaning the saddle, bit, and bridle.

In the field, grooming to make the horse look nice would often fall to the wayside when troopers were tasked with other, more important duties. However, practical care, like the removal of sweat from the horse's body, would be done regularly (if the horse sweats heavily and then is "put away wet", it can develop a number of negative health conditions).

Troopers might carry some oats or grain with them for the horses, but more than a day's supply would become prohibitively heavy- a cavalry horse, working hard, needs a minimum of 8 pounds of grain or oats daily, not counting the need for hay and water.

Which brings up the point of feeding and watering- A trooper would carry a muzzle bag or grain bucket, from which oats or grains could be fed, and he would carry a water bag, so that he could water his horse when necessary. Officers would be responsible for finding appropriate places to stop the cavalry regiment for resupply and feeding. Ideally, they would stop near a body of water, to allow for the easy watering of the horses (a horse can drink 5 gallons in about two minutes if they're thirsty), and with grass, to allow for the grazing of the horse (grazing is when the horse eats forages for grass either at liberty or with the trooper holding the horse while the horse is haltered). With that in mind, grass is often poor food for horses. It can cause indigestion, if the horse's stomach isn't accustomed to that grass, it takes a while for the horse's to chew, and it takes a while for the horse to eat enough to get enough calories to replenish from the amount of work that they undergo while on campaign.

Beyond that, a supply train would be absolutely necessary to bring along necessities like hay and grain. High quality hay is definitely the preferred meal for a horse that is in heavy work. The average cavalry horse (using that term liberally), would need around 20 pounds of hay. However, hay, being expensive, could be partially substituted for straw, which was also fed. So, the daily feeding of the horse would be something like 10 pounds of hay, 10 pounds of straw, and 8-10 pounds of grain or oats, with grass whenever possible and water whenever possible.

Farriers would take care of all other necessary horse care actions in the field. The farriers would put shoes on the horses, trim the horse's feet (if necessary), file down the teeth (when coupled with a bit, irregular wear on the teeth [which are constantly growing] can cause pain in the horse's mouth), sew up wounds, treat scrapes and gashes, and ensure that the horses are not suffering from things like indigestion, colic, exhaustion, or whatever else. When necessary, a farrier would also be tasked with shooting a horse with a broken leg. After a battle, farriers would go through the field and shoot the horses too injured to continue moving.

When the horse was stabled in times of peace, they would spend much of their day at liberty or in a stall, and they might be subject to a few hours work as a hunt horse or a carriage horse. Alternatively, they might be a farm horse, but draught horses were typically disliked for military use.

TL;DR: Horse care is complicated, heavy, and expensive.

Sources:

"The handy horse-book", by Maurice Hartland Mahon. While this text was published in 1867, the methods and practices of cavalry in the field remained largely the same throughout the bulk of English military history, and much knowledge on the topic was passed along by "word of mouth" until codified by the "Pony Club."

"Farriery Improved: Or, A Compleat Treatise Upon the Art of Farriery" by Henry Bracken, 1737, who was a fairly prolific writer on the topic of military farriery.