name change on birth certificate

What is the name of your state? Texas
I have a son who is 16, and a daughter who is 15. Their biological father has never had any part in their childhood. I have been married for 14 years, and they have always gone by my husbands last name. I have homeschooled them up to this point, but now they are going into public school. They would like to have their last name changed to my husband's name. What procedures must we take?

Petition the court for a name change.

I'm not sure I understand what you mean. Do you just go to the courthouse and fill out a petition to have the names changed? Is the petition signed by myself, the parent, or by the respective child? Thank you for your reply.