I think that one of the most overwhelming features revealed about racial relations in Reconstruction is that slavery, while abolished, was still in existence in a more surreptitious manner. While the 13th, 14th, and 15th Amendments were added to the Constitution, Southern states were quick to pass and enforce Jim Crow Laws which represented segregation, creating "two Americas" where one represented possibility and promise and the other reality and crushing denial of opportunity. This represents that race was still an issue which could not be easily overcome, confirmed by the Supreme Court in its decision of Plessy v. Ferguson, where practices of segregation were upheld as not violating the Constitution. In this light, people of color were confronted with how a nation can profess equality, enshrine in its Constitution how equality should be mandated, but still not come close to assuring it in practice.
While slavery had ended by 1865 amendment to the Constitution, it's safe to say that overall attitudes about black equality with white America hadn't shifted significantly, and the Reconstruction Era proved this.
There were exceptions, such as the volunteers in the Freedman's Bureau who contributed to the education and training of former slaves, but efforts such as theirs were not in the numbers large enough to make a real difference, or to indicate a trend or a change in American beliefs overall.
Unfortunately, Reconstruction is the time where white America, both North and South, abandoned the notion of black equality, or at least, failed to embrace it at all, and they would continue to suffer under heavy and blatant legal and social discrimination for the next one hundred years and, some say, beyond.
I am not really sure that it shows anything about the relationship between the two races today. But it might say something about the relationship back in those days.
I suppose it could tell you that in those days whites did not really respect blacks. The whites from the North were (you can argue) just using the blacks' votes to get themselves into power. The whites from the South hated the blacks and wanted to take away as much power as possible from them.
So the main conclusion there would be that whites did not really care about the blacks.
The Reconstruction Era demonstrated the reluctance of the whites to accept the changes that were supposed to occur following the abolishment of slavery. The government saw that the black people who had been cared for on farms would not longer have anyone to provide for them nor could they provide for themselves. They needed help from the government in order to make a new start in a country that had denied and still denied them basic human rights.
For blacks it meant a time of hope. Although freedom did not really seem to offer much, it still meant that they were no longer owned. However, it did not offer them any protection from people who chose to commit violent hate crimes against black people.
It was also a time of the expansion of schools for black people in the south. While there had been a few before, the Quakers and other groups from the North established some schools to help educate younger blacks. The literacy rate of black Americans increased during this period as they had formerly been denied the right to learn how to read.
White supremacist groups surfaced to terrorize black people in an effort to punish them for any progress that they seemed to make.