When Adam and Eve sinned, God judged them. Since they were the father and mother of all of mankind, and they were the only representatives of mankind at the time, all of mankind figuratively sinned in Adam and Eve. God's judgment was correct, because given the chance, every human has sinned.
What then happened to Adam and Eve? They were ushered out of the Garden, and God put a cherubim at its entrance to guard the Garden and the Tree of Life so that nobody could get back in. This is why at times the Bible bids people to return to God when they had never seemingly turned away from Him. Yet, all of mankind did turn away from God in Adam and Eve, and He invites us to return to the place, symbolically, where everything started, back to the environs He occupies, where the Tree of Life is.
The relationship with God is everything to our salvation. Without what Christ did in dying for our sins, we would not be in the position to have one with Him. Christ's payment of our sins opens up the way for a relationship to be built and for us to grow in the Holy Spirit, because now we have access to the Tree of Life in a relationship with God.