According to the Bible the book of Genesis and its events are not how we are to trust that the Bible is true. We accept those facts after we have come to know it is true because Hebrews 11:3 says it is by faith that we know the worlds were framed by the word of God. Do I expect someone who has no faith in the Bible to understand that fact? No.
Isaiah tells us what to look for to determine if there is a divine being, and how to know who that being is: "Remember the former things of old: for I am God, and there is none else; I am God, and there is none like me, declaring the end from the beginning, and from ancient times the things that are not yet done, saying, My counsel shall stand, and I will do all my pleasure." Isaiah 46:10. And this is reasonable. I can't predict the future. I can take a guess at it, but those guesses are limited to within a few hundred years at most. No one from the 16th century predicted that we would have the lives we do, or who would be president or king. We as men do not know those things that far in advance. Yet in the book of Daniel there is a prophecy that even to the critic spans over 2000 years. In the second, seventh, eighth, and eleventh chapters the history of nations from Daniel's time to ours is depicted. I believe the book was written in 6th century BC, but skeptics think it was written as late as 2nd century BC. It can be verified it existed by then because of the Septuagint (a Greek translation of the Old Testament with the apocrypha). If you read Daniel 2, you find that 5 eras are depicted. The reason I say eras rather than nations is because of the ten toes or the divided nations in the dream. The eras have come to be understood from the interpretation given in that chapter of Daniel to be first Babylon, then later on in chapter 7 and 8 and in the account of the fall of Babylon the Medes and Persians are understood to be the next kingdom, then Greece as it is named in chapter 7 though not named in chapter 2, Rome is understood throughout the chapters but never stated, and then a divided kingdom which bears some semblance to the empire that preceded it in that has the elements of Iron still present showing that the divided kingdom is basically divided Rome which we know as present day Europe. Only 5 eras are predicted before the everlasting kingdom comes in. Just to give the skeptic the benefit of argument, let us say Daniel was written at the beginning of the 2nd century BC because it needed some time to become canon to the Jews for it to have the position it does in the Septuagint. That said can you explain to me how only two other national periods have come after the Greek rule? The Assyrians had quite an empire, the Babylonians also, the Persians after them, the Greeks for some time, and then the Romans. Then Europe has remained as the fallout of the Roman Empire with no single figurehead from the day of its breakup until now. There have been at least three attempts at a single empire by Charlemagne, Napoleon and Hitler. Hitler and Napoleon almost dominated all of Europe, but despite being brilliant tacticians made some bad calls at just the right moment. Only 5 eras predicted and that prediction has held the test of time for over 2000 years. That is not by chance.
Now I realize that doesn't prove the whole Bible true, it just proves the book of Daniel is true. But Daniel had supernatural information he was writing that we know is accurate as you study that book more closely. Daniel in chapter 9 quotes Jeremiah as inspired when he tries to understand the content of his vision in chapter 8 and references things that Moses wrote about. So Daniel considered at least the book of Jeremiah and Deuteronomy as inspired by the same One who gave him supernatural information. You all should be smart enough to figure out the rest of the connections if you just read without jumping to conclusions immediately.
As for the supposed discrepancies in Genesis 1 and 2, I just want to point out that anyone who has ever read book, manual, specification at some point gets something confused as they read and sees a contradiction. Normally we go back and reread to figure it out. However, because of bias and a desire to brush off the Bible is why people see what they want to see. In any case Genesis 1 deals with what happened during the creation week with the first three verses of chapter 2. The second chapter is a more detailed narrative of the events of the sixth day. Just because God made man and woman on the same day doesn't mean He made them the exact same moment. And dealing with chapter 2 verse 5, please read verses 4 and 6 before you jump to the conclusion it was a whole other account. If you read the three verses together you realize that this is a summary of chapter 1. Dew was what watered the ground rather than rain. It did not rain at all until the flood.
Now for you Christians who don't take the Genesis account literally, how can you do that and be Christians? If Adam and Eve evolved, then death came before sin. But Paul in the book of Romans in chapter 5 says that sin brought death. Jesus also treats the account as literal when dealing with the question of divorce. Peter also talks about those that in the last days scoff at the creation account (2 Peter 3:3-5). The account of Genesis 3 is based on the literalness of the preceding chapters and is how we know we need a Savior by describing what is the mess we are in and we how got there. Otherwise if that is not true, then Christ is no Savior, Paul is a fanatic and the other apostles with him, and all you have is death and nothing more because if Jesus was sent to give us everlasting life, and yet before sin there was death, then why would God give for lawlessness what He didn't for perfect obedience. At least be consistent with the book, consistent with the Pope, or consistent as an atheist, but please be consistent.