The kind of doctors who want to help and heal should attend to how much is already right with the patient. Every Christian church is inherently Trinitarian as the people of God, the body of Christ, and the temple of the Holy Spirit; some of those churches need to hear about their Trinitarian foundation more often. Christians live by the grace of Jesus Christ, the love of God the Father, and the communion of the Holy Spirit (2 Cor 13:13); some of them need this benediction placed on them more emphatically. Every Christian prayer that goes up finds its way to God the Father because of the mediation of the Son and the intercession of the Spirit; pastors should draw attention to the direction of that current so that the people who pray to the Trinity can see what is always already Trinitarian in their prayers. Every soul that is saved has been adopted by the Father who sends the Spirit of his Son into our hearts crying “Abba, Father” (Gal 4:6).
And Ben Myers recently had a tweet storm on the subject. The first eighteen tweets are momented here (strictly speaking, it's eighteen plus a footnote). The only thing I would say differently (although I suppose I could go either way on Trinity Sunday) would be the handling of analogy; I would say something like:
Ancients: Analogies show that we aren't talking incoherently.
Moderns: Analogies are models of the Trinity.
When it comes to the Trinity, the Church Fathers develop analogies promiscuously but modestly and without putting very much weight on them; the Scholastics put more weight on the received analogies, but cautiously and modestly; but we moderns are all too often immodest about them. (Leading to the very misleading 'models of the Trinity' talk.) I've noted this about misreadings of Augustine's On the Trinity -- people will talk about Augustine's analogies as if they were intended to be direct descriptions, despite the fact that (1) the reason he talks about the analogies at all is to argue that we can make sense of the words used to formulate the doctrine of the Trinity; and (2) he spends a considerable portion of the last book emphasizing the ways in which even the best analogy fails.
Then he gives the principles underlying the doctrine (the tweets after the eighteenth have not yet been momented by anyone, so I'll give them here until they are):
#19. Basic principles of the doctrine of the trinity: (a) The canonical principle. OT and NT are a diverse but coherent witness to one God— Ben Myers (@FaithTheology) May 25, 2017
#20. (b) Creation principle. The one God is creator of all things and so is not on the same plane as anything else— Ben Myers (@FaithTheology) May 25, 2017
#21. The spirituality principle. God is spirit. God has no body and is not comprised of anything like a material substance— Ben Myers (@FaithTheology) May 25, 2017
#22. Simplicity principle. Because God is spirit, and because the creator transcends space & time, God must be indivisible and without parts— Ben Myers (@FaithTheology) May 26, 2017
#23. Abstraction principle. Words can be used to speak of God only if they are stripped of all connotations of space, time & matter— Ben Myers (@FaithTheology) May 26, 2017
#24. Revelation principle. How do we find the best words to use? We'd better stick to revelation. (These words still need abstraction)— Ben Myers (@FaithTheology) May 26, 2017
#25. Fitness principle. Language about God should be fitting to God's character. (Early Christians had a special term for this: theoprepes)— Ben Myers (@FaithTheology) May 26, 2017
#26. The fitness principle (25) comes solely from Greek philosophy and is by far the biggest piece of "hellenisation" in this doctrine— Ben Myers (@FaithTheology) May 26, 2017
#27. But this piece of Gk philosophy was used critically against the Gks: it was used to distinguish monotheism from pagan anthropomorphism— Ben Myers (@FaithTheology) May 26, 2017
#28. So there is a true hellenisation at this point (and really only at this point) but it is a critical and subversive use of Gk philosophy— Ben Myers (@FaithTheology) May 26, 2017
#29. So now that we've got the basic principles, next I'll talk about the content of the doctrine of the trinity— Ben Myers (@FaithTheology) May 26, 2017
The only thing I would add to the above would be the importance of baptism -- the baptismal formula keeps coming up, it's the fact that the Trinitarian heresies conflict with it that makes them so serious, and, of course, as far as we can tell the Creeds grow out of the liturgical practice of baptism -- the Symbolum Apostolorum being in its current form mostly a Frankish expansion of the Old Roman Symbol, which is a summary of the Roman baptismal declaration of faith, and the Nicene Creed being apparently an Eastern summary or summaries of unknown provenance (it used to be thought it originated from Caesarea, but this is less accepted now), with a similar function, that was (or were) modified by the first two ecumenical councils. The baptismal connection is also important for something Myers discusses a bit later, about the practical implications of the doctrine, and it is to some extent the root of what Sanders discusses in the article linked above.
Tweets 30-47 apply the above principles, and tweets 48-55 discuss the implications; I won't put them all here, but if someone moments them, I'll link to the Moment.
ADDED LATER: And Myers has put the entire tweet storm (up to 58, with the three final tweets wrapping up the whole) in a Moment.
ADDED LATER: Myers has put the whole thing in a blog post.