If I recall correctly, I started researching my Carothers genealogy before I graduated from high school. That was over 20 years ago. When I started, I knew my father's name, my Carothers grandfather's name, and my Carothers great-grandfather's name. But I had no information beyond that. So, I started with William Henry Carothers, my great-grandfather, and began looking for him in a US Census when he was a child, to find out who his parents were.
This was back in the microfilm days. I went with my Mom to the San Diego Family History Center, and we began looking through rolls of microfilm. Rolls and rolls and rolls. Finally, my Mom hit pay dirt! She found William Henry at the right age with his parents, Clark and Rachel Carothers. Much subsequent research over time has made firm that relationship.
I can't recall at the moment what information we had that led us to look through Missouri's US Census. We must have known his birth place was in Missouri. I'm guessing we must have used an index and were looking for Carothers families in particular counties in Missouri, but I can't recall that detail. I'll have to look through my old research notes.