We all know about Brown v. Board of Education, the Little Rock Nine, etc. However, I've long been curious about the pace of integration in the North. My white mother was good friends with a black classmate in Junior High School in 1961, but she can't remember if she had any black classmates before that in elementary school. The Internet is surprisingly unhelpful, even for major cities.
Oh hey I can actually answer this. Currently writing my thesis on race in Ohio after the Civil War and you may be surprised to find out that a lot of places had integrated schools. Cleveland for example was one of the most integrated places in the country. From what I have read, even before the Civil War most of the New England states had integrated schools. The former Northwest territories states were a bit different and I can't speak for Pennsylvania and New York. In Ohio black students were banned from attending white schools up until around the 1850s if my memory serves me right. Free blacks still had to pay taxes however, for schools their children couldn't attend.
States such as Ohio, Indiana, Illinois, and Michigan varied greatly on school integration. From my research I have found that quite a few places had segregated schools because black members of the community wanted them. Teaching was one of the few higher paying jobs that African Americans could get into and school integration would mean the loss of black school teachers.
Check out Reconstruction by Eric Foner, Black Ohio and the Color Line by Gerber and especially We Will Be Satisfied With Nothing Less by Hugh Davis. The last one specifically focuses on schools.
Now I can't tell you if any of this changes. I have only read up until the 1880s, so there could be segregation later.
One of the important things to note though is that this is a receding trend. Most of my sources are academic journals and if you like I can give the specific references later (on mobile) but Northern schools have been segregating over the last forty years. In fact schools in the North today are nearly as segregated as southern schools at the time of Brown v Board of Ed. This is also true of schools in the West, particularly in larger cities. Part of the issue was the lack of segregation decades ago, and no formal laws requiring integration (most of them applied to the south). So what you actually had was a mildly integrated North school system undergoing segregation at the time of southern desegregation.