What is the first of going through a private adoption, if both parents give up all rites to the unborn. What will I have to do first in order for me to take the baby out the hospital. I live in Washington DC. Please help with the steps that I need to take

You need an attorney. You need to make sure all the Ts are crossed and the Is dotted. Custody and guardianship do not enter into it. You might want to review this thread from a person encountering a similar situation: https://www.askmehelpdesk.com/family-law/mother-willing-give-up-rights-unborn-baby-me-634497.html

You need the adoption papers signed off by a court to take the child home.