If traveling by land, would travelers have found existing or well worn trails? Would there have been inns along the way where money could be exchanged for food and shelter, or would travelers be expected to forage for themselves and camp out along the way? How common was travel on foot relative to travel on horseback or by buckboard stages of that era?
My interest in this is because the fledgling Latter Day Saint movement envisioned a "New Jerusalem," which was to be settled in northwestern Missouri, near present day Independence. They prepared an expedition and left Kirtland, Ohio (near present day Cleveland) in the spring of 1831. They traveled the approximate 800 miles in about 90 days. I've read the "boots on the ground" report from a participant, Ezra Booth. The return trip was said to have been somewhat easier because they returned via the Missouri and Ohio rivers. Steamships made it possible to travel upstream. Still, the description of men walking paints a picture of hearty individuals. I am not sure why they didn't go on horseback. Any help or corrections will be appreciated!
Steamboats would have been the best; the burgeoning Southern cotton economy fueled a lot of steamboat traffic on the Mississippi river system, so it would be possible to travel from Pittsburgh to western MO via the Ohio, Mississippi, then Missouri rivers. George Catlin, the artist, would travel a long way into the west via steamboat in this period.
For traveling on foot, the National Road would also extend to Jefferson City, MO in the 1830's, and there would have been inns along the way.
The Mormons were the first to settle in the vicinity of Kansas City.
A group of Mormons from New York settled there in 1831, but they were forced out by hostile neighbors in 1833 and there was no further settlement in the Kansas City location until the establishment of "Westport" in 1834. The town was not named "Kansas" until 1850.
So, theoretically, it was impossible to travel from Pittsburgh to Kansas City in 1830, because Kansas City did not exist.
To travel to the future location of Kansas City would have been difficult, and the 90 day journey of your Ohio Mormons would have been typical (they would have been among the first settlers to travel there).
If they did not travel with horses, it was probably because they were too poor to afford them. After all, the Mormon handcart pioneers trekked west to Utah from 1856 to 1860. They didn't push all their belongings all that way on handcarts for fun. They were too poor to afford oxen or horses.