You are correct. In the Catholic Church, Baptism is the first sacrement that is received. They believe that this sacrement is nessessary in order to remove any remnants of the sin at the fall from an individual's soul. I think they view it like, at the fall, Adam and Eve were the only two humans in flesh form, but God already knew the souls of everyone who would ever receive a fleshly body on Earth. So when Adam Splashed for the first time in the puddle of sin, all the other souls were in close enough proximity to be splashed with a little sin mud as well.
So the first sacrement is designed to wash away any residual "original sin mud" from the soul and it's done at infancy so if the child dies, he/she can go into heaven with a clean slate.
I have way over simplified this and if I misrepresent the Catholic doctrine in my explanation, I sincerly apolgize.