This Sunday is the Solemnity of the Most Holy Trinity. This great feast fittingly takes place very year after Pentecost, the holy day commemorating the descent of the Holy Spirit and the birth of the Church--after the ascension of Christ, sent by the Father, comes the sending of the Spirit, and thus we see the Triune God, one God in three Persons, made manifest to us. It's an appropriate moment to stop and consider for a moment the most mysterious of the mysteries of our faith.
I titled this post "Approaching the Trinity" because it is downright foolishness to think one can comprehend the Trinity. To understand God in His inmost being, in which unity is three-ness and three-ness is oneness? You can't get your arms around it; the best you can do is walk up to it. It's very hard to say what the Trinity is, but a bit easier to say what it isn't. In this post I'll show you some of the boundaries of thought on the Trinity (this is why the other half of the post's title is "Avoiding the Extremes"): you can know then that, if you find yourself thinking in this way, you've gone too far and are no longer thinking of God as He is.
Now, if you're approaching a mystery in which you are trying to see in what way something can be both one and three, there are two obvious ways you can err: overemphasizing the oneness, or overemphasizing the three-ness. The first of these errors often becomes a sort of modalism, while the second tends to become tritheism.
Modalism is the idea that God really, truly in His being is one, but he only appears to us in different persons; that is, God appears to us in different modes. Under this framework, the Israelites would have encountered God in "Father mode," and then God would have incarnated in "Son mode," then been present to the Church in "Spirit mode." It seems nice and tidy, and avoids that messiness of trying to explain how one God can be three different Persons, and was the sort of thought that many an early heretic fell into (and not a few modern theologians, I'd wager). BUT this way of thinking doesn't fit our data from Scripture. When Christ speaks of the Father and the Spirit, he speaks as though they are different from him; yes, he says, "The Father and I are one," but he also says, "My Father and I will come and dwell with him," and "I will send you a Paraclete." Sometimes his language denotes unity, sometimes differentiation. What's the solution? Either there is, in some way, both unity and differentiation, or we would be forced to conclude that Christ is a liar, putting on some show to make us think there are three Persons involved when really there's just one. And since all parties would agree that Christ is God, and God does not lie, the last solution does not work. So modalism can't be true.
Tritheism is the idea that God really, truly is three different beings, is three gods, but that they are all one in willing the same thing, or something like that. I think that many people today tend to conceive of God in this way, that there are these three beings each of whom we call God, but we call them one God because they just seem to get along so well. Not only that, I think most people tend to become subordinationists, too, placing the Persons of the Trinity into different degrees or ranks, one being somehow higher than the other: when they think of God, they think of the Father as being really God, and then, oh yeah, the Son, he's pretty god-ish, too, and I guess the Spirit, we can't leave him out. Unfortunately, as with modalism, you could find scriptural support for such a position, as many early heretics did, simply by pointing to all those places where the Persons are spoken of as distinct: how can Christ pray to his Father if they are one being? Well, we could stray off into complex discussions of the Trinity sharing one act of existence, or the thornier questions of what exactly we mean by "God exists" if God isn't a thing among other things in the universe, but rather the ground and source of all that exists, and other such deep metaphysical topics that I'm not sure I understand myself, so instead I'll go with something a bit simpler. Anything that is to be called "God" must be infinite. There cannot be more than one infinite, because in order to be two, there would have to be something that was not the other thing, and thus neither one would really be infinite. So, if God is infinite, God must be one; hence, we cannot understand the Trinity to be three gods.
I would hope that this could be of help in your spiritual life. If you're thinking of the Triune God either as an actor with three masks, or three guys who are just super-chummy, then you're not really thinking of God at all. To help hammer these points home, take a look at the creed attributed to St. Athanasius of Alexandria, one of the great Fathers of the Church, who defended the divinity of Christ and the integrity of the Trinity against heretics of his day. Let this be a small start in coming to know God better as He is.