In order to gain shared custody of my son, would I have to prove the mother unfit?

In order to gain shared custody of my son, would I have to prove the mother unfit? The only reason she was awarded sole custody in the beginning is because I screwed up the dates and showed up to court a day late. Since then she has refused to let me see my son "because she can" (her words). I have a motion regarding custody all filled out, along with a factual document about how I am equipped to take care of my son. One on how it would be in his best interest to have his father in his life, and a parenting plan that starts off with a phasing in process due to lack of contact between him and I. I do not know what else I might need. I do not want to be laughed out of court.