Either a civil or a religious ceremony is acceptable. The couple is simply required to indicate that they intend to take each other as spouses and say words to that effect. After the ceremony is completed, the newly married couple files a marriage certificate with the proper state or local authorities and they are considered married. The person who performs the wedding frequently files this certificate for them within a few days of the wedding. Typically the couple and the person who performed the wedding sign the marriage certificate.
Some states require that the witnesses sign it as well.
Work with an experienced lawyer in your state.