1) When you're an illegal, underground movement, you don't generally commission sculptors or painters to depict your sacred stories or beliefs, lest that artist turn to the nearest centurion and shout, "Yo, another one for the lions over here!"
2) There was a bit of a hangover from Judaism with its strong prohibitions against making images of God or gods, making Christians wary to portray God.
3) Likewise, since they lived in a pagan world, and the pagans loved their statues and mosaics of the gods, Christians tended to associate such artwork with paganism, and wanted to distance themselves from it.
After the legalization of Christianity in the 4th century, great public churches were built and artwork began to increase: we see mosaics and paintings of the Trinity, of the saints, of scenes from Scripture. But in the 8th and 9th centuries a movement arose called iconoclasm (literally, "image smashing" - this what I meant by "smashing icons": not "Oh, excellent, well done, smashing icons, old chap!" but rather breaking icons into tiny bits). The iconoclasts had various motivations. Some said that any images of Christ, the Trinity, or the saints amounted to idolatry, the worship of images, strictly prohibited by the Scriptures. To make icons, they said, was to violate the First Commandment.
St. John Damascene made several arguments against this. First, very simply, because God had become man in Jesus Christ, God could be depicted, rendering the Old Testament prohibition against making images of God null. Second, the veneration of icons was an ancient tradition which had borne abundant spiritual fruit. And third, he stressed that the veneration shown to an icon is not directed to the image itself, but rather to the one whom the image depicts; when I venerate an icon of Christ, the image is serving as an occasion and a point of focus for my veneration of Christ himself. I'm not worshiping the image, but the one imaged. Thus St. John defended iconodulia (veneration of icons).
One group made a theological argument against making images of God, attacking Damascene's first point: they claimed that, because Jesus is a divine person, because he is God, and because God cannot be described or depicted, therefore we cannot depict Jesus. They acknowledged that Christ indeed had a human nature as well as a divine nature, but asserted that his human nature was one that could not be drawn (the ten-dollar word for this position is agraptodocetism, agrapto- meaning "cannot be drawn," -docetism meaning "seeming," as in "only seeming to have a fully human nature, with all a human's attributes"). Some even went so far as to say that Christ had all colors of hair, all possible heights, all possible noses, etc.!
Theologians like St. Theodore the Studite defended the full humanity of Christ, including its ability to be depicted, against these heretics. There is nothing essential to being human that Christ lacked, they argued, and that includes the ability to be described. The Incarnation means Christ became truly human, which includes having a particular hair color, height, etc. St. Theo argued, "You would have it that Christ became incarnate not into the world, but only into your minds, only as an idea." But because Verbum caro factum est, et habitavit in nobis ("The Word became flesh, and dwelt among us" - John 1:14), Christ can be drawn, and icons are legitimate.
Further, because Christ said, "He who has seen me has seen the Father" (John 14:9), it was argued that it was not inappropriate to portray God the Father in some way. (This was more prevalent in the Western Church; the Eastern Church still tends to be wary of imaging the Father.) Two modes of depiction seem to have become dominant.
The more common one was taken from the Book of Daniel, in which Daniel has a vision of God:
I beheld till thrones were placed, and the ancient of days sat: his garment was white as snow, and the hair of his head like clean wool: his throne like flames of fire: the wheels of it like a burning fire. (Daniel 7:9)This "Ancient of Days" is where we get the image of God the Father as an elderly man. And it just looks so much better to have a big flowing beard on an old man, so artists tended to add that on.
A less common mode, but one popular for a time, was to take John 14:9 very literally and show the Father as looking like Christ. If you've ever seen a religious painting with what appear to be two Christs and wondered, "What the Samuel F. Hill is that about?" that's what's going on. It's not Jesus' brother Jerry (Robin Williams's joke), or a high-class ad for Doublemint gum; nope, it's an artistic way of illustrating the idea conveyed in this passage of Scripture.
There ya go: a little history, theology, and art history all in one!