Many people have attempted to illustrate this concept using many different approaches, some are convincing, others like the Three-part egg theory is kinda far out. Fundamental theological perspective for mainstream Christianity; you must accept the Trinity or you cannot be Christain period.

  In the ancient world, we see a process that took place before arriving at the conclusion of the Trinity. Just to say; Jesus is God would imply different meanings in the second century, we have many Christologies explaining how Jesus could be God and man at the same time. 

   Docetism; Christ is God and he only seemed to be human, fully God and phantasm. We see this taught in the second-century gospel of Peter. If Christ is not human; then he could not spill his blood for the sins of mankind, so the doctrine was rejected over time.    

  Separationist View; Jesus and Christ were different, Christ comes from heaven and takes over Jesus body, Jesus was the "God-man," both God and man joined together while Jesus was alive, Jesus dies on the cross and Christ ascends back into heaven. This view lost out because the Christology teaches two aspects of the same person. 
   
  Modalism; I am a Father, also a Son, and Brother. Modalistic Christology became the standard view for a lot of Christians in the second century; three modes of the same God. God is the Father, Son, spirit; only one God not three and the Father was only eternal. The majority refused this theology around 380AD when the Trinity became orthodoxy, because of the concept "you can not be the son of whom you are the father of" the teaching was rejected over time. 

  Popes/Bishops in the second to the third century of Rome were Modalists and so the Trinity being presented unto them seemed to mold into their theological perspectives quite well. One God in three persons was a process of elimination for the orthodox Christians. Constantine wanted unity for his people, and he had to get Christians together as one unit. 
 
  Several thoughts of Christology never made it through orthodoxy, the Church needed an answer in the late middle of the third century and the Triune God became the compound conclusion. The first Council of Constantinople 381 C.E. is when Trinity became established to reject heresy. The gospels have different prospectives about who Jesus was; so creating the Trinity doctrine harmonized the gospels and the people at the time, and therefore became predominant.  

 Problem verses; Some gospels portray Jesus as fully God while others present him as fully man. How could God be God and Jesus be God at the same time? Does not that make two Gods? Many other believers in ancient world thought Jesus was not god at all, not divine just 100% man. Others thought Jesus was 100% God and not a man, therefore this created an issue. 

  Trinitarian groups sound good with Professors teaching the Trinity out of Johns Gospel, exegetically with Greek composition, centuries of trinitarian scholars editing and translating Greek texts. Critical analysis proves the earliest Christians had exaltation Christology, not incarnation Christology, look at the entire gospel of Mark.       

  Modern Non-Trinitarians attempt to harmonize the gospel canon conflict by altering the gospel of John; stripping Logos of deity, or making the Logos a created deity, bending contexts and clear meanings. Non-Trinitarians would do well if they would leave the gospel of John alone but most alter Greek and ancient religions, changing the intent of the narration.