The LORD God commanded the man, saying, "From any tree of the garden you may eat freely; but from the tree of the knowledge of good and evil you shall not eat, for in the day that you eat from it you will surely die."
The tree was indeed a tree of knowledge according to the Bible. However, it was not just any kind of knowledge -- it was, specifically, "knowledge of good and evil".
Adam and Eve were created in an initial state of almost-complete ignorance of good and evil. The Bible teaches that where the law has not been given, sin will not be imputed (Romans 5:13). Adam and Eve were originally sinless because they had virtually no command of law. In eating from the tree, they essentially received the command of law, at once becoming knowledgeable of good and evil. Now that there was a law, sin would be imputed.
Adam and Eve immediately began to conform their behavior to the law they had just demanded, covering their public nakedness (Genesis 3:7). However, the full force of the law was far more than they could bear -- more than anyone other than Christ has ever been able to bear (Acts 15:10).
According to scripture, the wages of sin is death (Romans 6:23). As a result of the coming of the law to Adam and Eve, their sin was now imputed to them, and ultimately, death was the result.
Part of the moral law that now governed Adam and Eve was the responsibility to train up their children in the way they should go (Genesis 18:19; Proverbs 23:13; Deuteronomy 6:7, 11:19; Psalm 78:5). They were morally obligated to teach their children the knowledge they had received through their disobedience. This means that their children were also under the law, and thus their children's sins would also be imputed -- their children would thus also die.
There is much, much more to the origin and transmission of sin throughout history than simply this, but I believe this presents a basic, Biblical introduction to the Bible's teachings on Adam, Eve, sin, and how it all relates to you and I.