Equal Rights for all during Reconstruction moved forward a bit, but there was still a long way to go. The 13th, 14th, and 15th Amendments were ratified during this time period.
The 13th Amendment: Abolished slavery
The 14th Amendment: Guarantees due process and equality under the law
The 15th Amendment: Right to vote (regardless of race...but not gender)
People called "Radical Republicans" championed forms of equality and helped enact some civil rights acts. They even placed federal troops in the south to protect minorities.
Even the president at the time was not happy with these changes (Andrew Johnson). He tried to veto many bills and was almost taken out of office by way of impeachment.
Once Hayes became president, many of the advancements went to the wayside. Federal troops were removed from the south, abd segregation (Jim Crow) laws came into effect.