Based on your question, I will assume that you are not married to the father of your child. My answer would be somewhat different if you are.
You would have to file a Petition for Change of Name for a Child with your local county district court. The process itself is not that extraordinary, although I do suggest hiring an attorney to guide you through the process.
The only concern in regards to the father is that if he still retains parental rights to the child (either naturally via biological link, or court-ordered as part of a divorce or in a stand alone Suit Affecting Child Parent Relationship), you would have to serve the father because the court would be an interested party in the suit.
Once he is served, it is up to him to try to prevent you from changing your child's last name, his only affirmative defense being that you are doing so for fraudulent purposes. Assuming you are not, or if he does not show up (which would be most likely the case), you should have no problem changing your child's last name.
If you do not know where the father is, the court will appoint an "ad litem" attorney whose job it would be to make sure that the father cannot be found, and that therefore court should serve him via a substitute service (usually publication). The only throwback in this scenario is that you would be responsible for the ad litem's fees - it typically runs anywhere from $500 to $1,500.
Good luck.Ask a similar question