I think the usual formulation of the Trinity is three coeternal persons (hypostases) sharing one divine nature (homoousion).

A good thing to keep in mind is that the Trinity is a mystery that's not fully revealed to us. What are the three's interactions, for example, at the time of Jesus' baptism on the Jordan river? We have the Father talking to the Son and the Holy Spirit descending, but why did they do it like that?