For this example think of God as a guy named Mike Smith. Mike is a father, a coach, and works in an office. When he is home his children call him dad and it is his job to take care of them. At his practices he is called coach Smith and his job is to make the team better. At work his name is mr. Smith and his job is to complete office work. God works the same way. He has three different names but these names all correspond to the same being.
As a former Christian turned agnostic atheist, I believe I can present a somewhat qualified opinion.
The three "people" in the Holy Trinity are different aspects of the same deity- or rather, three incarnations defined by different attributes. They are in spirit inextricably linked but appear to humans as three separate beings.
To reconcile worshiping Jesus, the son of God, Christians made him into God. This is both confusing, not truly monotheistic, and logically impossible. Jesus sometimes knows what God wants/is thinking and sometimes doesn't, depending on what the writers need at the time. Sometimes he does God's will and sometimes he questions it. God, Jesus, and the HS all seem to be able to operate fully independently of each other at times, and don't always seem to know what the others are thinking. This is not how one being operates.
One has to think theologically to begin with. We are talking about Christian doctrine, so we must use Christian Scripture.
It says there is one God. Only God may be worshiped. Note that this God refers to himself in plural.
This God creates things through his will, his words, and then operates in the created things through his spirit. All these things, have the same being as each other and therefore are one, yet they can materialize separately. (the Gospel of John sets up Jesus being God).
Then we get to the logical problem of three things being one thing.
1+1+1 does not equal 1, but 3. Therefore Christians must be illogical or polytheistic.
BUT, in that argument the 1's are not being put into unity, rather they are simply being put together. So, to put them into unity would be:
1x1x1 which, in fact, does equal 1.
So we can conclude that it is possible for three separate "ones" to be "one". And thus, the Trinity can logically make sense.
We just need to define our terms first. What do I mean by God? An entity that is eternal, all powerful, all present and all knowing. I would also describe this entity as possessing three persons. So the Trinity is one God that possesses three persons. Or put another way: One what, three whos. To claim that something doesn't make sense is to say it is contradictory or incoherent. It's not either one of those. Contradiction would be God is three persons and one person. And incoherent statement would be blue smells bad. The terms conflict.
Christian view of what God did:
First I will create two people with original sin,
Then I will insert myself into a women,
So that I can be born to die on the cross,
As a sacrifice to myself,
To make me forgive the people for the sins that I originally condemned them to do.
From a scientific standpoint, one person occupies one body. Having three people within one body is impossible. From a non scientific standpoint it still doesn't make sense. Essentially the trinity is saying that God, sent himself down to Earth, as his son (so God himself is God's son), to die on a cross so he could sacrifice himself to himself, so he could forgive sins that he created. Not logical at all.
The trinity was not made canon until the 4th Century. It seems to be based off the Mithras religion common in the Roman Empire. Blending cultures and beliefs of the local people made conversions easier and I am not against keeping one's culture. The issue I have is when the native religion begins to overtake the evangelized religion (in this case, Christianity). Is the burning bush a person of the trinity? It was the word of god, and the word was god. This means according to Trinitarians, the bush needs to be part of the Holy Quadernary. Personally, I believe their is absolutely one god, just as the jews and muslims believe. When god says "let us..." I think he is talking to angels. There is also the possibility that Elohim (plural - God) is like many languages do with politeness - the pronoun is put in the plural.
For starters the terms used to describe the trinity are not in scripture. That's the first sign that it isn't true.
Terms not in Bible but used to describe Trinity:
God the son
God the spirit
Three in one
One in three
Those are PRETTY key terms to the trinity that aren't in scripture.
Second off, the trinity is never explicitly stated. If you ask one to prove the trinity they have to hop all over the Bible to give evidence. This evidence usually isn't concrete though, for example common evidence is when baptizing you are baptized in the name of the father, son and spirit. But that never says the three are one, it just has the three in a verse.
Why is this an issue? Well lets look at the rest of the main doctrines we know are true.
God's character - whole chapter for it Exo 34
Jesus is God's son - ALL over the new testament
There will be a resurrection - whole chapter for it (1 Cor 15)
Jesus was raised from the dead - detailed accounts in the gospels
There is a kingdom - Ezekiel takes a multi chapter tour through it
My point is all the other major doctrines in the bible are clear as day. Yet the trinity is a "mystery" as most people say. (Really? Who God is, is a mystery?)
Let's continue though. There are many points where Jesus and God clearly aren't the same.
Contradictions of the trinity:
God's immortal Jesus died (at least 3 days)
God's all knowing Jesus had to learn the scriptures
God's all knowing Jesus didn't/possibly doesn't know when he is returning
2 separate wills (Garden before his death "your will not mine")
God cannot be tempted (James 1:13-15) Jesus was tempted everyway we are (Hebrews)
Jesus said his father is higher than him.
There's a lot more, but you get the point.
Trying to argue the spirit is God is VERY difficult. The best thing you get is sometimes similar words are used for the two, but the spirit is never called God or anything close to it.
Also the fact that the doctrine wasn't created until a few hundred years after Jesus's death is a big deal. Your telling me the Jews, God's chosen people, didn't know their God was a God of three parts? I know they weren't the best followers but it isn't in ANY of their writings both in and out of the Bible.
Overall- pretty clear it isn't a true or sensible doctrine
Via. Abductive inference, quantum superposition and the temporal impossibility of causality, we can infer that it is most likely that God doesn't exist. From a non-Christian point of view, theism itself is illogical. The Holy Trinity is the epitome of a theistic position. As theism is illogical as (a) causality is impossible, (b) causality is not required, (c) abduction reveals metaphysical naturalism is more likely, and (d) quantum superposition leaves an external cause impossible, the Holy Trinity is illogical and "doesn't make sense".