My daughter's legal name baby girl smith cause her mother never turned in any paper work at the hospital how can I get that changed to her actual name and put my name on the birth certificate?

It is not as simple as that. You must file a petition to establish paternity and within the petition seek a name change to add yoor sirname. As to her first name, you might need mom's consent. By the way, as to your sirname, it is not an automatic absolute right. I am a lawyer with 32 years experience in family law.

If you do a custody case the name change can be included and that will be free.

Yes the court must establish paternity, and then just make the request for the name change as you proceed with your case.