Well to start, keep in mind that anti-miscegenation laws were older than the sovereign United States and were intended to keep the races separate. Thus mixed race children created a problem here and had to be put in one of the two categories. For the time period you asked about (slavery era), free people of mixed race were considered legally white even if individuals had up to one-eighth or one-quarter African ancestry depending on the state. For most of this time, more important than exact genetic makeup were things like, which race one looked more like, community standing and association, etc. This is partially thanks to the frontier era making it unlikely that people had any documentation of ancestors. These are all subjective and tough to quantify traits so the experiences of different mixed race children were likely not all consistent with one another's.

Perhaps a more clear time period is the 20th century, primarily in the jim crowe era. The animosity between blacks and whites between the civil war and the next century evolved, though I would have trouble explaining much of that as I don't know much about it, but I digress. Once again we have a problem of differentiating between black and white, but this time we don't have quite the same level of difficulty with ancestral documentation. In comes the first one-drop rule in 1924. This means if you have "one drop" of black blood in you then you are black, and thus you get treated as such. Of course there were some people that had black blood looked white enough that they were able to get around the rule, but this is difficult to really quantify so I won't really comment on it other than that they likely had quite varied experiences based on a number of factors.

http://essays.backintyme.com/item/22

Joshua D. Rothman, Notorious in the Neighborhood: Sex and Families Across the Color Line in Virginia, 1787–1861

http://www.pbs.org/wgbh/pages/frontline/shows/jefferson/mixed/onedrop.html