A person is allowed to feed his pets on Shabbat. However, it is not permitted to take unnecessary measures to feed the animals on Shabbat. One may put fruits and vegetables before an animal without cutting it. However it would be forbidden to cut the fruit as this would be an unnecessary tirha (labor). Of course, if the animal cannot eat the food unless it is cut, like a gourd, it would be permitted.
It is also permitted to guide the animal to a place with grass, for grazing. However, if the grass was cut on Shabbat, we fear that a person may pick up the grass, which is mukse, and feed it to the animals on Shabbat. Therefore, in this case, one should not bring his animals to graze in a field with freshly cut grass.
Finally, just as we are to rest on Shabbat, so too our animals are meant to rest. However, they are allowed to do melachot on Shabbat. They may not, however, perform work for the owner on Shabbat. This is why they may eat from the uncut grass even though it would be forbidden for us to pull the grass out.
