Contradictions
There are several contradictory ideas within the Bible. I'm going to try to list some here.
Do not sin/ask for forgiveness of sin
Predestination and Free will
The Bible speaks about predestination, that He chooses who would become conformed to the image of His Son (Romans 8:29-30) according to the good pleasure of His will (Ephesians 1:3-6). Paul sees himself as being set apart before he was born, called through His grace, to have His Son revealed to him (Galatians 1:15-16).
We know there is no man that seeks after God (Psalm 14:2–3). We are dependent upon God's mercy to save us (Romans 9:14-16). Then why are those God did not choose sent to Hell? If it is up to the Potter (Romans 9:19-21), then is it their fault God didn't choose them?
I am grateful for the ones God has chosen, that He has chosen any at all! That He has shown his Mercy and Grace. So then if God has picked His flock, why does the Bible ask us to choose Him (John 3:16)? If God has only picked some, how come we are told He doesn't want anyone to perish (2 Peter 3:9)?
The way I look at this is that from God's perspective He has chosen who will be glorified and who will be set apart for destruction. From our perspective, we do not know who He has called. It is our responsibility to pick Him.
The Bible says the devil put it into the heart of Judas to betray Christ (John 13:2). Jesus Himself knew he would do this and didn't stop him (John 6:64). In fact, Christ being delivered over for death on the cross was God's predetermined plan (Acts 2:23). Yet blame is put upon Judas the individual (Matthew 26:24).
Therefore it is on each one of us to either choose to follow Christ or not, regardless of the fact that Christ has already picked who will follow Him. I love how John 6:37 says "Everything that the Father gives Me will come to Me," which I feel describes predestination, and then it says "and the one who comes to Me I certainly will not cast out." which I think of as free will. So we see that "no man seeks after God" but He also says if we seek we will find (Luke 11:9)! That is why I personally cry out "Lord I believe, help my unbelief!" (Mark 9:24)
I do not understand why God has set things up this way and find it rather confusing, but I take great comfort in knowing He is in charge and I trust His design because I know He is a Good, Merciful, and Just Father.
The Trinity: three Persons in One
What is not here
Please note I have not listed Creation/Evolution as conflicting ideas that still both exist. We can see micro-evolution, but the start of all things began with God.