I live in the state of Ohio
I had a son out of wedlock 2 1/2 years ago, the biological father did not sign the birth certificate. I have full custodial rights. Although the father still does see him occasionally, He does not pay child support and isn't active in our lives.
My son is starting to learn things like his name and where he lives.. and it's a little confusing! How much would I have to go through to have his last name changed to mine? Or, where would I even start, contact wise?

Does your son have the father's last name on the birth certificate?

At any rate, you need to have an attorney file the name change paperwork for you in family court for the name change.

To change his name you will need to get the fathers permission. Why is he not paying child support, when you went to court and got custody did you not get a child support order

A birth certificate is a legal document. Therefore, you will need to go to court to have it changed. It is highly likely the father will need to give permission for the change.

Go down to your local Family Court and inquire about the process and requirements.

The father is not on the birth certificate, nor did he sign.
My son does have his father's last name on the birth certificate.
He is ordered to pay child support but does not.. He has been struggling with addiction for the past couple years and has not kept a job.

I know that He won't agree to let my son's name be changed if given a choice, but He is barely in our lives and isn't steady. I'm the one who on a day to day basis has to schedule appointments and sign for things.. I'm the only support my son has, so I want him to have my name.

Again, since the birth certificate is a legal document it requires a court order to change it. So you need to go to your local Family Court as I suggested. If the court requires the father to sign, then that's what's needed. Its not going to matter how much a part of your life he is or isn't.

And if the father refuses?

Then the court won't allow it.