How? With what money? With what resources? With what education? You're talking about an entire population that was intentionally deprived of familial connections, cultural connections, the ability to organize, the ability to build wealth, the ability to exercise any autonomy, literacy, and education.

Africa is not exactly a small place, and most ex-slaves didn't even know where their ancestors had been kidnapped from.

Also keep in mind how much different things looked at the end of the Civil War than much later. Ex-slaves were promised equality with whites, full rights as citizens of the US, and given the promise of reparations for slavery. Congress passed a law in 1865 that guaranteed full citizenship regardless of race and the 14th amendment was circulated starting in 1866 and became part of the constitution in 1868. For a decade following the end of the Civil War Reconstruction proceeded at a fast pace. Laws were changed, progress was made, historical iniquities were being redressed. The vast majority of ex-slaves in this situation who were offered the possibility of staying wherever they were and using the labor skills they already had to attempt to make a living in America (either through sharecropping or on their own) seemed enormously enticing.

At a minimum the situation looked to be superior to their previous situation of enslavement. They were ostensibly free. They could keep their families together, they could build their lives up (in terms of wealth, community, education, skills, ambition, etc.), and they had the prospect of attaining true equality of stature and accomplishment with whites in perhaps a generation or so.

It was not until two or three decades later when Reconstruction had been destroyed and dismantled, when slavery had been replaced with a racial caste system that was becoming enshrined in custom and law (Jim Crow et al), and when it became abundantly clear that the end of slavery did not mean the end of white supremacy in America that black Americans began to comprehend that the society they lived in was going to limit the extent of their advancements to a very narrowly defined box not much expanded from where it had been before. And then there really was a huge debate on what to do. Black communities felt the oppression, understood the long-term implications and generally understood that the status quo was untenable.

Eventually they did take action and move, out of the South and into the North and the West in one of the most significant demographic shifts in the 20th century called The Great Migration. By then they had more money, more resources, more education, much greater literacy, and greater ability to move around (due to the advent of automobiles and the advancement of railroads). But even so, and even moving within the US alone, it was an enormously challenging endeavor that not all African-Americans undertook.

If you want to get some additional perspective on what things were like I'd suggest reading "The Warmth of Other Suns" by Isabel Wilkerson.

Sorry for the brief drive-by; I can expand more on these if you'd like

Didn't they want to start over again somewhere free from all that persecution

Yes, and this explains migration to different parts of the US from the South, such as Harlem in New York City, other urban areas such as Chicago, and even lands on the frontier -- the newly (as of the late 1800s) settled areas in the midwest and west

Also didn't the US government thought it would be wise to help them start over somewhere else.

Somewhat; this is long before emancipation, though.

There were two lines of thinking in the US at the time: 1. that the nation could not integrate large numbers of free blacks, and 2. free blacks would not enjoy the full blessings of liberty in the US. The group behind the eventual colonisation of Liberia was known as the American Colonization Society

Prominent Americans such as Henry Clay, Daniel Webster, and John Randolph were among the best known members of ACS. Former President Thomas Jefferson publicly supported the organization’s goals, and President James Madison arranged public funding for the Society. The motives for joining the society were vast as a range of people from abolitionists to slaveholders counted themselves members. On the other hand, many abolitionists, both black and white, ultimately rejected the notion that it was impossible for the races to integrate and therefore did not support the idea of an African-American colony in Africa.

...

The U.S. Government had provided Liberia some financial support, but Washington expected Monrovia to move toward self-sufficiency. Commerce was the first economic sector to grow in the colony. However, French and British traders continually encroached upon Liberian territory. As it was not a sovereign state, it was hard-pressed to defend its economic interests. The U.S. Government lent some diplomatic support, but Britain and France had territories in West Africa and were better poised to act. As a result, in 1847, Liberia declared independence from the American Colonization Society in order to establish a sovereign state and create its own laws governing commerce.

Here's your source, a pretty easy read from the State Department