They keep saying crossing the red river, then heading west to cross the Brazos. If you cross the red river, you're way past the Brazos. By 1883, the Chisholm Trail, now I-35, was running 5,000,000 head of cattle, towns, ferries, and supply posts were all along the route to the Kansas railroads.. They should be surrounded by cattle runs.
The square piano that Elsa plays is almost entirely in tune. A piano of that era, and that construction would have been wildly out of time from the constant jostling of the wagon, the change in temperature and humidity. It's also very unlikely the piano survived significant damage from being unloaded in haste. The piano should have sounded at least discordant but nothing like as was shown.
While educating the immigrants before their long journey, he warns them about poison ivy and poison oak. He holds up a piece of the plant to show to the immigrants. The greenery he shows them is actually from a White Oak tree.
The group of German and Eastern European immigrants goes in the span of very short time in the first episode from having just one person who barely speaks English to several people who are conversant or at least understand. There has been no schooling or contact with civilization during the short time span.
Elsa plays Moonlight Sonata by Ludwig van Beethoven in 1801, composed well before her birth, not "Clair du Lune" (French for Light of the Moon) by Claude Debussy, which was not written till 1905, as previously asserted.
There are two plastic water bottles visible in the back of a wagon at the 2:47 mark of the show.
At the 38:54 mark when Josef is translating for Shea and speaking with another immigrant, the screen reads [speaking German]. They are speaking Russian.
Josef tells Shea that it's illegal to swim where he is from and that they whip the bodies of the drowned before they are buried. While this was true in Germany in the 1530s, it was not the case in the 1880s, 350 years later.