How exactly would the average person have emigrated from the South to Colorado 1865-1870 and what would have been the most common path?

In 1865-1870, the Smoky Hill Trail was the most popular route. This went from Atchison, Kansas, to Denver. The Smoky Hill Trail saw wealthy people taking the stage (Bayard Taylor did in 1866), merchants taking wagon trains, the poor taking handcarts/wheelbarrows, and the very poor just... walking.

The Smoky Hill Trail had plenty of dangers. Eastern Colorado was an active war zone. Especially after the Sand Creek Massacre of 1864, Native Americans increasingly lost any hope that the United States would negotiate in good faith. Why try to work within treaties that are discarded and broken? Why trust in peaceful negotiations when a peaceful band under the protection of the US was slaughtered even more brutally than those engaged in active combat with US troops? The Dog Soldiers swiftly became unaffiliated with the Cheyenne. They were already moving that way in 1864, but now they were a radicalized multinational army in their own right, composed of just about any Native or part-Native who wanted to go down fighting. The Dog Soldiers attacked Fort Rankin (now Fort Sedgwick) in 1865 and shut down Julesburg and the traffic flow from the Oregon-California Trail. They also massacred the inhabitants at the American Ranch post office in 1865. These are just the major skirmishes, and the newspapers constantly reported attacks - some of which were made up to sell papers, to be fair.

Throughout the late 1860s the Dog Soldiers (which sometimes included the adult Bent children; George Bent was himself a former Confederate soldier, like many Native Americans) fought battles at Beecher Island in 1868 and Summit Springs in 1869. These both are not along the Smoky Hill Trail, but are close enough to it to be concerned.

The relentless attacks and patrols of the Dog Soldiers also meant that the Smoky Hill Trail and the stops along it could change from year to year, and maybe even quicker than that. For those walking the trail, their slow speed made them vulnerable and sometimes meant they followed the wrong trail and died. Depending on the railroad crews slowly making their way across Kansas was better, as they had security; but the railroad didn’t reach Colorado until 1869. Once it started, though, railroads used the Smoky Hill route quickly. The transcontinental railroad was finally joined in Comanche Crossing (now Strasburg) in 1870. (Promontory Point was a publicity stunt; rail travelers still had to disembark and board a ferry.)

Why not a more southern route that connects with the Santa Fe Trail and then diverts north to Denver? After all, it had more of a connection to the south. But this would require cutting across Texas, which was still also the territory of Comancheria. These two nations would fight over the direct southern route to the West until the mid-1870s. The major power keeping the peace in Western Comancheria had been the Bent family for decades, but they’d ceased to be an influence and now the Bent children were mainly opposed to the army and any reconciliation with the US.

There was an option if you absolutely had to go via Texas: The Chisholm Trail hooked Texas up with Kansas. That still means going via Smoky Hill from Kansas rather than taking the Santa Fe. In the 1870s, as Comancheria and the nations within it were rounded up and forced into reservations or killed, the Texas-Montana Trail became a much more direct route through the Colorado plains. The Battle of Washita River was fought in 1868 against the Southern Cheyenne/Arapaho tribes under Black Kettle, and that was the beginning of the end. While Black Kettle was not Comanche, Comancheria was no longer something of a haven for other nations, and the US domination of Texas became inevitable. This was great news for the cattle trails, not so much the tribes.