Adam was not the first man, if we can't grasp this simple truth then the rest gets pretty much a superstitious mix of fairytale and dogma. We know beyond doubt that humans have existed for tens-of-thousands of years. Adam is a generic symbolic form of each human and his / her journey through their spiritual life.
So now that we have established that Adam was not literally the first man, then what might be meant by, first man and sin entering the world?
A religion in the age of Adam began to teach that G.d was One. Such a teaching suggested that humanity had reached a milestone and was not symbolically human in a spiritual sense and thus the first man had arisen in a spiritual sense from all the human beliefs that had gone before.
As for sin entering the world. We each have a dual nature, human and animal. The human nature is goodly, The animal nature is selfish, self-centred, greedy, etc. These are the traits we label as sins. The animal nature and its associated sins is at the core of our being. This is what we must overcome, as you mentioned, to be better humans. So sin did not enter the world through Adam, sin is an inherited state of having evolved through nature.
