I do not want to sound “New Ageish,” but wrapped up in the idea of the Trinity is that God is a relational energy that charges all that is. God is not an abstract being that stands aloof to us. God is Father*, Son, and Spirit in an internal relationship that spills over into all of humanity. The created world is Trinitarian at its core in its necessity and yearning for relationships. This is how it bears the mark of its Creator and how our very being bears the image of God (Genesis 1:27). Therefore the mere mention of God is to speak of relationship and to say the name of any person is to speak of relationship. This relational quality was indicated by Jesus when he said that the two most important commands are to love God and love one’s neighbor (Matthew 22:36-40). This boils down God’s purposes to relationships.
You may be thinking that we should scrap the Trinity and stick to the teachings of Jesus to make this point. This would make things easier to understand and eliminate any potential debate about three gods. But if we did this, relationships may be perceived as preferred rather than essential. The Trinity presses us further. God is a relational being that has woven the essence of relationships into the fabric of creation. Relationships are the stuff of life.
*God is not confined by gender, and there are biblical images of God that are feminine. I use it here because it is continues in the language of our liturgy, but I am not implying that God is male.