Hi Jeremy,

Understanding the trinity is no small thing. It took the church almost 400 years (and several heresies) to be able to come to an agreement as to what is meant by God - the Father, Son and Holy Spirit. So this answer may take some rereadng to fully understand. I would also encourage you to read and reflect on the bible passages.

The Jews wanted to stone Jesus because he ‘being a man’, claimed he was God (John 10:33).  Jesus doesn’t dispute that he is a man, and both he and the Jews are talking about God as a third party.  He says he is the Son of this God, they say that’s blasphemy.  Jesus has just said, ‘I and the Father are one’ (in John 10:30).  But he doesn’t say ‘I am the Father.’ 

The Father is in the Son, and the Son is in the Father (John 10:38), but they relate to each other - they are not ‘the same’.  John 1:1 says, ‘The Word was with God’ (distinction), ‘and the Word was God’.  We can’t say, ‘The Son is the Father’, but we can say ‘the Son and the Father are One’.  They are not one Son, or one Father, but One God.  They relate to each other by the Holy Spirit. 

They relate eternally and perfectly - they are always relating, and are never separate.  The three ‘persons’ of the Trinity (the ‘threeness’) are not separate.  But they are always Father, Son and Spirit, because they are always relating as the Father, the Son and the Spirit.  So although the three persons are not separate, they are distinct.  They are not ‘the same’.

They never act separately - For example, God is active in creation, Father, Son and Holy Spirit. 
(For references, see:
Genesis 1:1, Deuteronomy 32:6 and Malachi 2:10 for the Father;
Colossians 1:15-17 or Psalm 33:6 for the Son (or the Word);
and Psalm 33:6 or Psalm 104:30 for the Holy Spirit. )
But they act distinctly.  For instance when the Son becomes a man, in the ‘incarnation’, the Father sends the Son in the power of the Spirit.  The Son is incarnate, the Father isn’t, the Spirit isn’t.  But we can still say ‘God is incarnate’.

Plus, we never see/hear/interact with one separately - you’re relating to them all.  When you have God’s Holy Spirit living in you, he is the Spirit of the Father and the Son (‘God’ and ‘Christ’ in the words that Romans 8:9 uses), so the Father and the Son live in you (John 14:23).  Jesus Christ, God the Son, is in you by His Spirit, and God the Father is in you by His Spirit.  The same Spirit, but again, Jesus is not the Father. 

By the way, that was why the Old Testament prophets could say “The LORD says”: because God lived in them and spoke through them.  It means we can trust their words, and Jesus’ words - in fact the whole message of the Bible - as being the word of God, not just of human beings (see 1 Thessalonians 2:13).

Hope this helps.