This is not exhaustive and it is not in order, and it is potentially not even correct. But after reading around a few things, here are my suggestions on 3 reasons why the trinity needs more airtime in our thoughts and life.
"They are inseparable and cannot part from one another, but keep to their separate courses within one another, without coalescing or mingling, but cleaving to each other... And there is one and the same motion: for there is one impulse and one motion of the three subsistences, which is not to be observed in any created nature."
God's Trinitarian being is not like water and it is most certainly not like an egg. The three persons are inseparable but totally distinct. It is beyond comparing to anything in creation. Which is great! Because without the trinity we end up with a being who is conceivable and imaginable. And He typically ends up looking like a man with a big white beard. But with the trinity, It means God remains a wondrous mystery, an object of true worship that we can stare at in awe. We can almost understand Him, we can genuinely experience them, we can truly glimpse the reality of God, but we can never fully comprehend or understand him. With the trinity, God loses his beard.
God is "unto himself all-sufficient, not standing in need of any creature which he hath made, nor deriving any glory from them, but only manifesting his own glory in, by, unto, and upon them..."
"In this divine and infinite Being there are three subsistences, the Father, the Word or Son, and Holy Spirit, of one substance, power, and eternity, each having the whole divine essence, yet the essence undivided"
The traditional confessions are always very clear on the sovereignty and all-sufficiency of God, and the Bible is littered with proof texts for it. [Psa 148:13] But the confessions (as in the 1689 LBC above) often describe God's self-sufficiency quite separately from His Trinitarian nature. I find, however, that it is precisely because God is '3 in 1' that his self-sufficiency makes sense. Without the trinity we have a sovereign, all sufficient God who is totally alone and would need to create something in order to experience relationship, or in order to truly experience love. But with the trinity we have a God who has eternally experienced relationship and love within himself. The Spirit has always related to the Son, the Son has always loved the father, and so on. This Trinitarian God truly needs nothing else outside of himself, not even for relationship. He does not need our attention or devotion, because his trinitarian inter-relationship is enough for Him. Which means he is free to voluntarily 'want' us. He is never a needy God. He offers relationship to us freely, and invites us into the Love that he has known since before creation. [John 15:9]
"I will ask the Father, and he will give you another Helper, to be with you forever…
Anyone who loves me will be loved by my Father, and I will love him and manifest myself to him…
The Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you…"
Whenever you describe God as a singularity you end up using adjectives. He is holy, he is patient. That's fine. But the beauty of the trinity is that you often find the most suitable way to describe it is to explain the relationship between the three and you find yourself using verbs. The son asks... The father gives... The spirit helps, teaches, reminds (see quote above). Without the trinity we have a God who is static and theoretical. But with the trinity, we have a God that moves and we can feel the movement. We can see the trail left behind their constant activity.