My ex boyfriend signed the birth certificate in Indiana but we moved to Tampa Fl. So do we share custody or do I have full custody? And can I give him visitation rights on my terms and can I say that I don't want my son around his new girlfriend?

You have what is known as defacto custody because the child us with you and there is no legal arrangement for custody. Legally, you both have the same custody rights. So, just like you currently have your son, and can dictate what happens, he can do the same were he to have physical custody of the child. It is in everyone's best interest to establish a court order governing custody and visitation.

NO, you have no say in what he does with his child during his time. If he is doing something damaging or dangerous, you call the police and take him back to court to modify the order. If he were to take you to court to petition for custody and visitation, and tells the court that you have been keeping the child away from him because of his girlfriend or because you want "visitation on your terms", you'll come out looking very bad in court. You could even lose primary custody. Your best bet here is to use the court system to protect your custodial rights, his right to see his son, and your son's right to have both parents involved in his life.