There are at least two ways to understand what the first sin was.
One way is to say that the first sin was disobedience to God. God had told Adam and Eve not to eat of the fruit of the one particular tree. When they did, they disobeyed him. There is support for this view of the first sin in verses 11 and 17 where God emphasizes that he had commanded them not to eat from that tree. This implies that what they did wrong was to disobey God.
Another way to look at this, though, is to say that Adam and Eve's sin was the desire to be like God. The serpent tempts Eve by saying that the fruit will make her like God and that is (according to verse 6) part of the allure of the fruit. Looked at like this, the sin is one of trying to be more than they are supposed to be and trying to think for themselves.