Well, for a start it's not a "teaching" since nothing is actually being learnt by the recipient of this "education" without first an obligation to skip over a whole bunch of rather wild logic-defying assumptions before they can even contemplate the question. And then all that is being "learnt" is that a monotheistic religion has developed a device whereby three distinct characterisations of deity can be accommodated with separate identities, while still retaining a united entity (without which of course it wouldn't be a monotheistic religion at all, which in turn would expose it as the scam it is, which is a bad thing for religionists who have problems entertaining the notion that they're as much the victims of te scam as scammers themselves, etc etc).

So there is really nothing to refute - the religious person can call it theology if he or she wants, but when they engage in semantic somersaults in order to get their heads around the fact that their religion's founders decided to compound the fallacy of inventing one mythical being by then inventing two more with contradictory characters, all they are really doing is providing evidence (if such a thing is required) that religion might make a big thing about narrow morals and rules, but it has never extended such a compunction to intelligence.

Should you wish to counter someone guffing about the "mystery" of their three-headed god you can simply tell them that there's no mystery at all in how they ended up with one, at least historically. Christianity was a jewish sect (hence the paternalistic super-deity) which elevated one of that sect's alleged spokesmen to deity himself even though the philosophy attributed to him was a radical one in the strict meaning of the term (hence the rather badly drawn messiah in the equation), and then popularised itself through blatant absorption of more esoteric views of deity already prevalent at the time, especially ones popular to Greeks (hence the inspirational spirit-type god thrown in for good measure). In doing so they ended up with what was being seen by more and more raw recruits as a three-god pantheon, and therefore invented the notion of trinity to make heretics of those who had gone too far in making such an understandable assumption. They had it in particularly for the church leader Arius at the Council of Nicaea who had been encouraging his own considerable number of recruits to believe just such an assumption. His notion of a hierarchical deity with the jewish guy on top - and indeed his nose allegedly - both received a crushing blow at the council from those who realised that the less jewish this thing was the better it would sell. Arius was sent packing and his recruits told to accept the new "mystery" or face the usual threats these guys like to dish out.

Just another day at the office for religious con-merchants keeping a good scam going, in other words. But since all three elements of their trinity are figments of the imagination it's no great worry at all to rational people. It's a bind the con-merchants made for themselves and it's one that no amount of theological pretensions will ever grace with the status of a philosophical or metaphysical debate, despite the religionists' insistence that it is such a thing. It's nonsenical bullshit, employed to explain nonsensical bullshit.

Don't worry about it.

 I always like asking Christians to cite the passage in the Bible that explains the Trinity.

 Seconded.  Don't get bogged down with them in theology.  These people, if they feel confident enough to rumble with an atheist, have likely already spent some time dissecting the beliefs of the Witnesses and Mormons and other groups that reject the Trinity and will just take their learning in this area to drag you down to territory that's familiar to them.