Goats don't eat grass as much as people think. They eat weeds, honeysuckle, brush, bark and so on. They aren't grazers like cows or sheep, they're "browsers" like deer.
(I don't have goats yet but I will in a couple of weeks. I've read enough today about goats that I can probably write a book. LOL)
Oh here's a quote from fiascofarms.com. That site is a must for anyone wanting goats or has goats.
Although the goat's digestive system is similar to that of other ruminants, such as cattle and sheep, who are "grazers" and eat grass, goats are more related to deer, who are "browsers". As browsers, goats are designed to eat, and prefer, brush and trees more than grass. It is natural for them to nibble a little here, and nibble a little there. Though goats will eat grass, if you are considering getting goats to be lawnmowers, you are going to be sorely disappointed, because they will eat your trees and roses before they will work on the lawn. They really like bark and will strip the bark off trees. (especially pines, cedars and maples, to name a few). Goats could be used to help reclaim grasslands that have been overgrown with brush. Our land was overrun with brambles, wild roses, honeysuckle and 100s of small pine tree when we moved here; these are all gone now.