Apart from main streets in towns, and separate from railroads used for long-distance travel, what would the experience be like traveling from, as an example, St. Louis to Kansas City with horseback riding and covered wagons being the main modes of transportation?
Were there designated roads, or just large areas of land for travelers to cross?
Were there service stations or any equivalent to modern-day gas stations?
Would it have been common to pass other travelers or have other interactions common on modern-day highways?
It would have been pretty terrible, frankly. Even in the 1920s and 1930s a long-distance journey in an automobile would have been anything but comfortable. There would have been designated roads, but before the 1920s, no standardized traffic signals. Very few service stations because there were so few automobiles to service. One reason why the Model T Ford really took off was the standardization in parts, making it much simpler to fix even in far off service stations. It would have been relatively common to see other travelers, especially after the 1910s. There start to open up National Parks, amenities aimed at tourists, and gradually - better roads. Many years earlier, like in the colonial era, for example, a traveler would have needed to exchange money every so often because there wasn't even a nationalized currency. This is all to say traveling long distances gets easier over time in the United States.
Its sort of a myth that Eisenhower drew inspiration for the The Federal-Aid Highway Act of 1956 from his experience in Germany during and after WWII. In reality, Eisenhower was well aware this country had a problem with poor roads. When he was in the service before the war, Eisenhower was part of a terribly uncomfortable convoy going from PA to CA. He needed little convincing himself we needed new roadways and it helped further stimulate the economy.
Long established Native American footpaths and trails across the landscape were also followed by the early trappers and settlers, and some of those evolved into wagon roads along popular desire lines. There was no need for service stations, but routes across the arid West were chosen for access to water and grass as well as easy grades and good footing. The coming of railroads meant there was little need for slow, difficult cross-country travel, and wagon roads fell into disuse in some areas, though short regional feeder roads arose to supplement the railroad network. As settlement expanded and farm- and ranchland replaced open prairie, the familiar gridiron of section-line roads was laid down in theory, though of course most of those remained unimproved until widespread auto ownership in the 20th century.