What rights did free African Americans in the 1860's have?

In theory, though not always in practice, and certainly not in the South, free African Americans after 1865 were granted the full rights of citizenship under the 14th Amendment, and the right to vote under the 15th Amendment. The 14th Amendment extended to all people born in the United States the full rights of citizenship, including the right to own property. Under Reconstruction, the Union Army used its military occupation of the southern states to enforce this new rule of law, and as a result, for the first time, African Americans became members of the House of Representatives, members of municipal governments, and began to exercise their franchise.African Americans in the northerns states had these same rights, and continued to benefit from them.

Unfortunately, once Reconstruction in the South ended, southern state governments passed the Black Codes, which introduced the poll tax, literacy test and other laws that made it practically impossible for African Americans to vote or hold office. Furthermore, the Black Codes gave municipal governments the right to issue or deny work permits for all manner of professions and skilled trades, and because these governments again came under the authority of racist whites, they shut African Americans in the South out of virtually all good paying jobs. This development forced most African Americans to either flee to the North, or take up sharecropping, a watered down form of servitude. 

So, for a short period of time after the Civil War ended and up until the U.S. military ended its occupation of the southern states in The Compromise of 1877, freed African Americans enjoyed economic and political freedoms that they would not again enjoy until after the Civil Rights movement in the 1960s. This period of history demonstrates that even when a government grants rights to its citizens, if that government is not willing to protect the rights of those citizens, then those so-called rights do not really exist.