William Tecumseh Sherman, the Commanding General of the United States Army in 1869, remarked of occupation duty, "Our soldiers hate this kind of duty terribly, and not one of our officers but would prefer to go to the plains against the Indians rather than encounter a street mob, a serve a civil process." This is a pretty good idea of the way duty in the occupied south was viewed. It was tedious, and if not, there was a good chance you'd be put to use either as a riot policeman or a armed, uniformed officer of the court. However, we should be careful not to go too far here, as complaints about service in the peacetime army were extremely similar, even in forts in the peaceful north, and especially in lonely western posts.

The Post-bellum US Army

We should also talk about another element of your question: the idea that someone might join the US Army in 1868 out of a sense of patriotic duty or lost adventure. It's possible that people joined the army for those reasons, but if so they would be vastly in the minority. Service in the army during the Civil War was dressed in trappings of fervor for the Union cause, and volunteers were viewed as heroes who sacrificed health and well-being for a worthy cause, but service in the regular army during peacetime was not viewed in the same way. If someone enlisted as a regular, it was generally because they had no better options. The regular army was kept small, its pay was poor, and the duty was dull. Bored soldiers misbehaved in garrison and in town, and one of the chief duties of NCOs and local civilian authorities was in tracking down and retrieving deserters, and returning stolen US Army property.

If your hypothetical, patriotically driven young man threw caution to the wind, damned their family and friends (who would likely condemn this course of action), and joined up anyway, they would be in for a very rude awakening. We have many, many records of men who immediately and continuously got in trouble in garrison posts because of boredom, resistance to arbitrary military authority, and clear dissatisfaction with army life. Even in the middle of the Civil War, when the Army of the Potomac was at the height of its morale, fresh off recent victories, and well fed, soldiers grumbled. Being in the army would have been terribly difficult, extremely boring, often confusing, and always subject to the whims and ego of officers of unpredictable competence. The same held very true after the war, when the cause had been fulfilled, the rebels thrown down, and the need for men in blue coats to march around was gone. All that was left was serving as an occupation force.

The demobilization of the US Army was fairly rapid. While at the end of the Civil War, the total number of federal soldiers was more than a million, the vast majority of those were volunteers, who were legally and practically distinct from the regulars, whose wartime numbers went up to no more than around 70,000. By 1867, there were 56,815 regular soldiers in the United States Army, and those numbers steadily dwindled over the next decade.

A large portion of those men would have been stationed in the occupied south. In 1868, out of 50,916 total regulars, 17,657 men served in the south. There's a good chance when you enlisted, you would be sent to serve in a southern state. How you got there is, I think, also worth discussing.

Recruitment for the army in the late 19th century is different from today. You would recruit directly into a regiment. Regiments employed recruiting officers in large cities - New York, especially - and recruited men directly into, say, the 23rd US Infantry. You would talk to an officer of that regiment and you would be shipped to that regiment to serve, directly, without dealing with too much in between. And so recruitment would likely try to sell you on the positives of whatever post that regiment was currently assigned. It might be a hard sell for a young, enthusiastic patriot to join up and travel all the way to sunny Alabama, to make sure men make their court dates and patrol the roads for Confederate bushwhackers. Not exactly something that encourages a soulful hurrah. I can almost imagine the recruiting officer, aware that this young man doesn't come from poverty, isn't a newly arrived immigrant, doesn't come from a profession that recently got outmoded, and would easily have comfortable prospects in a variety of civilian professions, raising an incredulous eyebrow and asking, "are you sure about this, son?"

Duties

The Military Reconstruction Acts determined the US Army's continued presence in southern states. At the highest political level, the Reconstruction Acts established martial law, and set the requirements for a state's reentry to the Union. Most importantly, these states had to ratify the 14th Amendment and grant voting rights to black men. While early idea of reconstruction included support for quite substantial redistribution of property, President Andrew Johnson, a southern Democrat whose political career was mostly in slaveholding Tennessee, kneecapped several elements of the initial program. The overwhelmingly radical Republican congress vetoed Johnson's attempts to override reconstruction, and maneuvered against efforts of sympathetic southern courts to get reconstruction declared as unconstitutional.

By 1868, Tennessee, North and South Carolina, Florida, Alabama, Louisiana and Arkansas had been readmitted to the Union. Georgia, Texas, Mississippi and Virginia wouldn't rejoin until 1870. What this meant for serving soldiers in the United States Army was that the guaranteed rights of citizenship, especially for black men, had to be protected. The declaration and sustenance of martial law in these areas also eliminated one of the tensions of using soldiers as, essentially, police officers or officers of the court that existed elsewhere; the military was the legal authority under martial law. Soldiers were put to work in building roads and railroads (engineering was, in fact, a huge component of military education at West Point) apprehending criminals, in seizing property (if applicable), of ensuring peace during elections at polling places, and other duties that resemble modern police work. Sometimes this was a specific part of reconstruction duty, in 1870, the Texas State Police were formed and empowered to police racial crime in that state, and employed black officers as well as former rebels to do so.

In fact, a good deal of what we might consider "police" work today was done by private detective agencies and railroad agents, many of whom actually worked for the Union army as part of the Bureau of Military Intelligence for the Army of the Potomac. The recruitment and payment of private citizens as semi-official police officers for the purpose of recovering army property and deserters was a fairly common resource the army drew on, even into the 1870s. Wild Bill Hickok served in this capacity in 1865, for instance, before taking several different positions as a town peace officer.

Your duties might also involve more dramatic events, such as policing feuds and combating racial terrorists. In Pulaski, Tennessee, systematic racial violence was under way as early as 1866. Beatings, shootings, murders, and riots were common responses to perceived "insults" suffered by whites from black men. One man, Allen Abernathy, was shot by a James Scruggs because Abernathy was whistling. A riot started after a black teacher whipped a white student for an offense in class. The latter example led to a white mob of "roughs," as newspapers tended to call them, being confronted by an organized group of black citizens who forced the roughs to back down. Some of these events coincided with or included members of the early Ku Klux Klan, but that group was, at this point, largely incoherent and scattershot in its organization and political goals. Still, anti-black violence was common, and a good deal of organized criminality kept a sort of postconfederate flair. The James gang of outlaws, for instance, were initially organized by members of Quantrill's raiders from Kansas and Missouri. Though they operated mostly in areas not under reconstruction, they are a fairly emblematic model of political criminals that existed in the postbellum years.

As a soldier, your ability to intervene in these events would be relatively limited, but, certainly, the US Army did intervene in riots, and employ violence or the threat of violence against white citizens in response to violence or threats against blacks. While that all sounds terribly dramatic, the bulk of your duties, even in a relatively violent place, would likely be the same tedious round of neverending chores that constitute garrison duty. Drill, fatigue duty of a dozen different varieties, parades and ceremonies would make up your day-to-day experience. On a lighter note, forts and garrisons often formed base ball teams that played against local civilian teams, and by the later 1870s and early 80s might also form rifle teams to similarly compete against civilians.

In short, service in the army of reconstruction would have been, for the most part, extremely dull. Many would rather serve on the plains in the Indian Wars, and still others might risk desertion to free themselves from service in the army. Soldiers in the south had to deal with anti-black riots, confederate criminals, and systematic racial terrorism. They also had to just get on with it and maintain their fort and garrison, keep the fires going and the kitchen stocked, drill and drill and drill. National politics would influence this role within your unit and outside it. It wouldn't be altogether different from service elsewhere, though, and because of prevailing political ideas, Americans by and large were resistant to military authority, and acted out when and if they could.