"The Father and I are one" is from John, and the meaning is found in John 17. It does not mean "The Father and I are parts of a triune God". John's Jesus, like the Jesuses of Paul, Mark, Matthew and Luke, denies that he's God, and makes it clear that he's simply God's envoy eg
John 1:18 No one has ever seen God; the only Son, who is in the bosom of the Father, he has made him known.
John 5:19 “the Son can do nothing of his own accord, but only what he sees the Father doing”
John 5:30 “I can do nothing on my own authority; [...] I seek not my own will but the will of him who sent me.”
John 6:38 “For I have come down from heaven, not to do my own will, but the will of him who sent me”
John 8:42 “I proceeded and came forth from God; I came not of my own accord, but he sent me.”
John 10:29 “My Father [...] is greater than all”.
John 14:10 “The words that I say to you I do not speak on my own authority; but the Father who dwells in me does his works.”
John 14:28 You heard me say to you, ‘I go away, and I will come to you.’ If you loved me, you would have rejoiced, because I go to the Father; for the Father is greater than I.
John 16:23 In that day you will ask nothing of me. Truly, truly, I say to you, if you ask anything of the Father, he will give it to you in my name.
John 17:3 “And this is eternal life, that they know thee the only true God, and Jesus Christ whom thou hast sent.”
John 20:17 “I am ascending to my Father and your Father, to my God and your God.”
And as to being 'one' with God, in John 17, Jesus says ─
20 “I do not pray for these only, but also for those who believe in me through their word, 21 that they may all be one; even as thou, Father, art in me, and I in thee, that they also may be in us, so that the world may believe that thou hast sent me. 22 The glory which thou hast given me I have given to them, that they may be one even as we are one, 23 I in them and thou in me, that they may become perfectly one, so that the world may know that thou hast sent me and hast loved them even as thou hast loved me.
In other words, says John’s author, Jesus' oneness is of a kind available to all believers, not an equality with Yahweh.