There was much in way of dreams and expectations that each group of people possessed at the end of the Civil War. Victorious Northerners set their sights on being able to expand businesses into the destroyed South. There is much in way of evidence to suggest that the largest driving force behind the industrial North was to be able to spread their sphere of influence into the South, allowing business to expand into new markets. Southern Whites had a more challenging predicament in that they had to rebuild their homes and settings, and had to end up reassessing their own state of being in the world. This helped to create a crater of disillusionment that cradled much of the South. Questions such as how the war wound up so decisively against them, wondering as to where their own personal wealth disappeared, and how to reconcile a social order that had freed Blacks rank along side Whites all helped to create a sense of consciousness in the world with many a question. Perhaps, this is why Jim Crow laws and practices of segregation began to emerge so quickly after the Civil War, one way of bringing back some order to a world that certainly was perceived to have lacked it. This questioning of identity certainly expanded to newly freed Black Americans. The Civil War's emancipation from slavery had allowed the achievement of freedom, or a life without slavery to emerge. Yet, many Black Americans had little idea with what to do with this newly granted freedom. The idea of having been in bondage all their lives and then suddenly being released into the world with no training, no preparation, nor transition was startling to many of them. Some ended up going back to their plantations, only to find they were not accepted. Think about how this would impact goals and dreams: The life of abuse and subjugation, the only life one knows, rejects and shuns. Even one's own cursed existence is still an existence and to experience rejection from that had to have been unimaginably brutal. Many Blacks migrated to the North, trying to find some way of life that could support them or a family. Others remained in the South, trying to do the same. The transcendent question of what would happen to Black Americans, freed in the wake of the Civil War, was something that would end up being addressed by thinkers like Washington and DuBois, individuals who proposed differing approaches to the Black predicament in post- Civil War America.
Of course, people within these groups did not all share the same aspirations. However, you can probably safely generalize, especially about freed slaves and Southern whites.
The freed slaves wanted "normal" lives in which they were their own bosses. They wanted black communities that would be stable. They also wanted land since that was the most obvious way of being self-sufficient.
The white Southerners generally wanted a return to self-rule. They wanted things to be (as much as possible) like they had been before the war. This is why the earliest post-war governments implemented things like the Black Codes.
The Northern whites are much less monolithic. Some just wanted to get on with their lives. Others, like the Radical Republicans, really wanted to remake the South so that it would be more similar to the North both economically and culturally.