As far as Confederate flags go, which today are a big part of the symbolism, for quite a while after the Civil War they were not allowed to be displayed. Immediately after the war they were treated as contraband and seized. It was illegal to keep such things, at least in the immediate aftermath of the war. During Reconstruction it was a little more complicated.

According to John Coski, while there was no explicit law forbidding display of Confederate flags during Reconstruction “Southern civilians presumably understood—or were made to understand—that flags associated with their late nation were forbidden in public."

In the years following the Civil War display of the flag, or of any other emblems of the Confederacy, such as uniforms, swords, sashes, buttons, etc, were seen by some as essentially treasonous acts (I don't think statues really existed until much later). And until the early 20th century, Southerners mostly kept their Confederate emblems stored away, even while resisting Reconstruction in other ways. The original Ku Klux Klan, for example, did not use the Confederate flag.

While there were a few exceptions here and there, by and large the display of the Confederate flag began to be used in a memorial and war-dead sense around the 1870s. Reconstruction ended in the 1870s and the use of Confederate flags and other emblems took off immediately. For example, in the 1870s some of the remains of Confederate soldiers who had died at Gettysburg were transferred to a Southern cemetary, draped in Confederate flags. This “memorial”, or “ritual” use of Confederate emblems was seen as generally acceptable, and blossomed between the 1880s and 1920s or so.

In the decades after Reconstruction, especially the 1890s, when the Confederate soldier generation reached its apex of power, the “Lost Cause” narrative took greater hold, and national organizations like the United Confederate Veterans (UCV, founded 1889), the United Daughters of the Confederacy (founded 1894), and United Sons of the Confederacy (1896) flourished. The UCV reached its apex around 1903, after which the Sons and Daughters kind of took over as leaders of the memorialization.

All of these groups placed especial emphasis on the Confederate flags, especially the battle flag. But they constrained their use mostly to memorializations, and events like Memorial Day/Decoration Day. A great emphasis was placed on putting little Confederate flags on the graves of the war dead. Still, it wasn’t until 1916 that the placing of such flags on graves became an official part of the UDC Memorial Day program. At the same time Northerners were increasingly doing the same, but with US flags.

In other words, for a decade or two after the Civil War Confederate emblems were basically banned from public display. When such emblems were used in public it usually became newsworthy and controversial, and people could end up in jail. A few decades more saw the rise of memorializing the war dead and veterans with Confederate emblems. Also around this time some memorializing monuments were built in the South.

Some of this was seen as a good thing in the North, in as much as it might help heal the wounds of the war and bring the nation together again. Unlike Nazi Germany, this was a Civil War, and there seemed some good in reconciling old differences through compromise. Over time veterans from both sides would meet at key battlefields, decked out in emblems of each side’s military, and shake hands, affirming renewed friendship.

After the war all Confederate flags were considered contraband, yet many were saved, hidden away in attics for decades. By the turn of the century it began to be acceptable to take these out and show them. Many people on both sides were amazed by their survival and the touching stories of how they had been saved by some brave flag bearer.

By the early 20th century this kind of thing had expanded into many forms of memorialization, such as statues, but such things were still almost entirely memorial.

Of course, in the early 20th century Confederate veterans and their children had come to control public culture in the South. It was also around this time that the “Confederate flag”, which previously could have meant many things, began to solidify around the battle flag, which eventually morphed into the rectangular “Confederate flag” common today. Interestingly, it was also around this time that Northerners began placing increasing symbolic weight upon the US flag. Laws were passed requiring the US flag to appear at all schools, and to protect it from desecration, and to honor it with “Flag Day”.

Anyway, it wasn’t really until around World War 2 that the Confederate flag, often in rectangular form, started to take on the role it still has today: One standing not so much for memorialization of Civil War dead and veterans, but of the Lost Cause, and often used in extremely kitschy ways. By this time the Civil War veterans were dead or very old. Generations had passed. It was generally felt that display of the Confederate flag was “a harmless and rather amusing gesture”. The Sons and Daughters of the Confederacy were, and still are I think, largely opposed to the use of Confederate emblems in anything other than solemn, memorializing ways.

Of course, since WW2 opinion has shifted quite a bit on the topic.

Most of this is from John Coski’s The Confederate Battle Flag. edit: tpyos

While only touching on what you ask here, you may find this response about Civil War Commemoration to be of interest, as it deals in part with how the narrative of the Civil War was shaped as a tool for reconcilliation.