In 1869, by wagon, southwest through Arkansas, probably in large part via the Southwest Trail. Several parts of my family made the same trek. By 1930, though, they almost certainly would have traveled by automobile. And it varies by personal circumstance: a single adult or middle-class couple recently married might well move to Texas by taking a train, while a poor farm family seldom would. A young man getting out of the service in 1945 might have ridden an intercity bus all the way from wherever he was mustered out.

In 1872, the Missouri Pacific railroad from St. Louis reached Texarkana, and was extended to Dallas in 1873. But a farm family relocating would still have been more likely to travel by wagon, bringing with them some treasured possessions, useful equipment, and at least one draft animal. Travel by riverboat from St. Louis down the Mississippi to New Orleans, thence by steamer to Galveston, was also possible but much less likely. Timetables for trains and boats (but not fares, alas) can be researched in Official Guides.

Those eight decades saw a lot of changes. Maybe you have more specific questions based on where in Missouri or Texas they were or the exact year?