Symbolism, Imagery, Allegory
When Father Latour stays with a nice Mexican family, he can't help but think of how interesting goats are, symbolically. For starters, they're usually considered to be pagan symbols, since the pagans liked to sacrifice goats in order to please the gods. You also have figures like Pan and satyrs—ancient deities that are half-man, half-goat, and 100% all about the ladeez.
Today we make an association between goats and horniness, and we can thank Pan and the satyrs for this weird, weird association.
At the same time, Christians have counted on goats for millennia to give them fleece and milk. Latour thinks of all this stuff when he looks at a goat:
The young Bishop smiled at this mixed theology. But though the goat had always been the symbol of pagan lewdness, he told himself that their fleece had warmed many a good Christian, and their rich milk nourished sickly children. (1.2.18)
In this sense, the goat marks a sort of transition between the strangeness and familiarity that Latour feels in New Mexico. On the one hand, he feels that this is a strange, pagan land. But on the other hand, it's a place in which he can do God's work, and this (like goat's milk) is nourishing to him. This is a contradiction Latour will have to overcome as he works to bring Catholicism to New Mexico.