God is made up of three person in one substance, correct? Are the three person separate and distinct forms? Are the three part separate manifestations that have distinct but separate relationships with the believer?
Yes, pretty much.
In mainstream Christian doctrine, the three persons aren’t ‘parts’ of the Godhead, they are each the whole of the Godhead, simultaneously - don’t worry, it isn’t supposed to make sense.
And yes, different relationships - Christians generally believe that God the Holy Spirit dwells within them and empowers them, while God the Son mediates in their relationship with God the Father.
So were the Father and the Holy Spirit also crucified since all parts of God are together simultaneously?
Has the Trinity been around since the beginning?
Only the Son actually experienced the Crucifixion fully, the Father & Spirit experienced it through association with the Son (in full empathic relationship with Him.)
and yes, in catholic/orthodox Christian doctrine, God has been Trinity from eternity past.
It isn’t three Gods that are in close communion.
It isn’t one God divided into three parts or aspects.
It is three persons, all of whom exist simultaneously but are the same being in entirety.
(at least according to Christian doctrine, anyway) - as I said, it isn’t supposed to make sense.
Christians like to point at things like a passage in Genesis where God says “let us make man in our image”, but the concept is foreign to Judaism, indicating perhaps that they read it differently.