XUSCR,
I figured someone would throw down this gauntlet. I apologise for my lack of structure in my response, but you have asked quite a broad question.
I will do my best to utilize only one of the tools that we as flawed and sinful humans have, the Bible. My other tools include prayer, faith, and my steadfast beliefs that are based on faith, research, and council. I will try not use Papal edicts, quotes from Catholic Certified Saints (unless said Saints are also authors in the Bible), or "tradition" as explinations.
The Trinity: Two of the more explicitly Trinitarian passages are Matthew 28:18-19 and John 1:1-14. In the first passage, the Lord said: “All authority in heaven and on earth has been given to Me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.” Notice that Christ uses the singular form “name,” not the plural “names,” when He gives this directive. This usage implies the unity of the Three Divine Persons in the Trinity.
John 1:1-14 says, “In the beginning was the Word (Christ), and the Word was with God and the Word was God.... And the Word became flesh and dwelt among us, full of grace and truth; we have beheld His glory, glory as of the only Son from the Father.” Paul echoed this theme that Christ is true God, the Second Person of the Trinity, when he wrote that He is “the image of the invisible God” (Col 1:15) and the “radiance of the glory of God and the very stamp of His nature” (Heb 1:3).
Simply, I believe in the Trinity because Jesus said so. "...Father and of the Son and of the Holy Spirit.”
Incarnation: John 1:3 "Who were born, not of blood or of the will of the flesh or the will of man, but of God." [In several ancient Latin manuscripts this reads: "who was born not of blood or the will of the flesh or the will of man, but of God."] John 6:42 "They were saying, "Is not this Jesus, the son of Joseph, whose father and mother we know? How can he now say, `I have come down from heaven?'"
"who was born... ...but of God." It's in the Bible. Plain and simple.
Here is a list of verses that pretty much say the same thing:
http://wbsa.logos.com/article.asp?id=4364 What's next? Oh, Virgin Birth:
"And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS. Then said Mary unto the angel, How shall this be, seeing I know not a man?" (Lu. 1:31, 34).
"But after he had considered this, an angel of the Lord appeared to him in a dream and said, “Joseph son of David, do not be afraid to take Mary home as your wife, because what is conceived in her is from the Holy Spirit. She will give birth to a son, and you are to give him the name Jesus, because he will save his people from their sins.”" (Matt. 1:18).
"But he had no union with her until she gave birth to a son. And he gave him the name Jesus."
(Matt. 1:25)
How can all this happen? God said so. He made the world and existance, so He can change the rules and governing laws of nature, physics, reason, etc.
Matthew ties it all together with "All this took place to fulfill what the Lord had said through the prophet: “The virgin will be with child and will give birth to a son, and they will call him Immanuel”–which means, “God with us.” (Matt 1:22).
God said so. Simple.