What political and moral gains did African Americans make during the Reconstruction and the years following?

The gains for African-Americans in the Reconstruction Era were huge. The obvious one is that slavery was ended, and millions of African-Americans who had been enslaved were freed.

Much of the African-American population moved out of the South, or tried to reunited with family members they had been separated from due to slavery. Some traveled hundreds of miles in order to do so.

But the gains went much further than that; for the first time in US history, African-American men were given the right to vote and hold office---and several African-American men were elected to local and state offices in several states. African-American families were allowed to send their children to school for the first time, and did so in droves. Literacy among African-American children skyrocketed from almost zero to near parity with White children.

This new enfranchisement also gave the Republican Party a large number of seats (remember, Lincoln's Republican Party was actually the liberal party at the time), and also led to the formation of uniquely African-American churches, mostly variants of Baptist and Methodist.

For a time, it really seemed as though White and African-American people might finally be able to live together in harmony.

Sadly, this was not to be, at least not yet. The new laws in Reconstruction, along with the Civil War itself, fomented resentment among many White people in the South, resulting in the formation of various White Supremacist organizations including the Klu Klux Klan.

Eventually the Reconstruction laws collapsed, and were replaced with a series of discriminatory laws called the Jim Crow laws. Racist White leaders were unable to completely roll back the clock---slavery remained illegal, for instance---but they did manage to suppress the political representation of African-Americans and generally make life awful for them once again. Racial segregation began during this period, and would not be eliminated by law until the 1960s. (De facto, it arguably has not been eliminated today!)