For everybody who is not married

Okay so I am a FTM and my boyfriend and I are confused as to what goes on when the baby is born. I obviously want him to sign the birth certificate and I want the baby to have his last name also. But since we are not married, is the state (New York State) going to go after him for child support and all that bs? We currently live together and plan on getting married eventually. We are on separate insurances obviously. I just don't know how this all works!