What can I do to receive custody of my son, even though me and his mother never married and I just moved out of the state?

Venue, in most states, follows the child.   If the child lives in Texas, then you would file a custody suit in Texas called a "Suit affecting parent child relationship."  Other states call the same proceeding a paternity suit or custody suit.  Regardless of the title, the general purpose is to have someone officially declared "mom" and "dad", and to set up a custody plan.  You are not required to have an attorney, but it is advisable to hire a family law attorney-- especially since this involves a potentially contested custody dispute.  If you are looking for custody, not just visitation, make sure that you have a plan laid out on how you are going to be a good parent to the child should you be awarded custody.  You want to demonstrate to the judge that you should be appointed as the managing conservator because you are willing to commit the time and resources to being a good parent.  Judges like to hear your plans for education, upbringing, and extra-curricular activites.  If you do hire an attorney, find one in the state where the suit will need to be filed-- which is the state where your son lives.  If you live out of state, you may need to save up some extra funds to travel back and forth for court hearings.