If you are looking for dogma..going line by line....that might be tuff but here is some resources I can provide you to reach a conclusion on your own.
If you go to biblegateway.com and type in the word trinity on the KJV (king james version the most popular version) and on the NIV (most modern intreptation of the Bible)
you get nothing either..as a matter of fact no matter which version u use you get nothing.
the dictionary reference has this to say about it.
"Trinity a word not found in Scripture, but used to express the doctrine of the unity of
God as subsisting in three distinct Persons. This word is derived from the Gr.
trias, first used by Theophilus (A.D. 168-183), or from the Lat. trinitas,
first used by Tertullian (A.D. 220), to express this doctrine. The propositions
involved in the doctrine are these: 1. That God is one, and that there is but
one God (Deut. 6:4; 1 Kings 8:60; Isa. 44:6; Mark 12:29, 32; John 10:30). 2.
That the Father is a distinct divine Person (hypostasis, subsistentia, persona,
suppositum intellectuale), distinct from the Son and the Holy Spirit. 3. That
Jesus Christ was truly God, and yet was a Person distinct from the Father and
the Holy Spirit. 4. That the Holy Spirit is also a distinct divine Person."
http://dictionary.reference.com/search?q=Trinity
Source: Easton's 1897 Bible Dictionary
I read an article by JH Keathley a pastor of 28 years and he says that the word trinity it not in the Bible.
http://www.bible.org/page.asp?page_id=215
The Jews written of in the torah were polytheistic. The first chapter gives credit to Gods not God for creation of the universe.
http://www.davidicke.net/religiousfrauds/christianity/piousfraud.html
In the above link there is proof upon translation that Moses and the other Jews were polythiestic in nature. At one points the new testiment they are shown worshipping other gods including a calf. At this point Moses gets the 10 commandments and the first one is in effect, "You shall have no other God's before me." This suggests again that there was more than one God in the piture.
Where Christanity and Judism diverge is after the old testiment which the Jews do not call the old testiment but the Torah and Talmund. We know though that in those books with Christanity has adopted as it own that polytheism existed in belief by the early Jews.
In the New Testiment the Holy Ghost is not mentioned by translation but instead added by the KJV (king james version).
The inspiration is not a new idea. In Hinduism the trinity is known as :
1) brahma =creation
2) vishnu= preservation
3) shiva = destruction
In life the Hindu's see everything as born preserved for a set time and than dying.
If you look at the Christian parallels you see a similar thing:
God=life
holy ghost=persrevation (in an afterlife)
Jesus =death as he was sacrificed for sins of mankind.
As to how the Catholics define it (they are the big propogators of the trinity theory), they have a really really wordy description here
http://www.newadvent.org/cathen/15047a.htm#I
They are quick to point out in that link in part 4 that the trinity is a mystery.
Ironically tomarrow markes the anniversary of the death of Michael Servetus who was burned alive at the state on Oct 27 1553 for suggesting the was problems with the Christian theory of the trinity he wrote a book about it called, "On errors of the trinity".
There are two societies that exist that debate his theories and their links are given below:
http://www.servetus.org/
http://www.miguelservet.org/servetus/web.php
Did you take your Bible to your church and ask? If so what did the pastor have to say?