What does the Church teach about the Trinity?
Let me try to explain, simply, what the Church teaches on the Trinity vis-à-vis monotheism.
The Trinity’s source is the Father. The Son is “begotten” from Him — not created by Him, “for there is nothing whatever that generates its own existence” (St. Augustine of Hippo, On the Trinity, Book I) — and the Holy Spirit is “the Spirit both of the Father and of the Son”, “proceeds from” the Two, and is “co-equal” to Them.
All must “honor the Son even as they honor the Father” (John 5:23) and recognize the Holy Spirit as “the voice of the Lord” (Isaiah 6:8-10, Acts 28:25-27). All are key to the “I Am”, the Logos, the principle on which everything — including reason itself — depends.
Philosophically speaking, because the Persons are so intricately connected (in mind, substance, action, etc.), They are One. This unity is to be mimicked in both the Church and sacramental marriage.
However, They clearly act distinctly (but not separately). This makes most sense, I think, in the context of the Eucharist: Father as being sacrificed to, Son as Sacrifice, Holy Spirit as the Person Who inspires sacrifice. They each have a place in the communitarian model.
They all must be God, though, because only God has these authorities. They must all be of equal power, also. How else could the Son be our Lord (John 20:28) and perfectly alike to the Father (John 5:19)? How else could the Holy Spirit bring forth the Son to earth (Matthew 1:18), how else could lying to the Holy Spirit equate to lying to God (Acts 5:3-4), and how else could our bodies belong to this same Spirit (1 Corinthians 6:19-20)?
That, in a nutshell, is how Catholics look at the Trinity — One, but Three; Three, but One. Thus, we chant, “holy, holy, holy” (Isaiah 6:3).
(cf. Nicene Creed, Athanasian Creed)