A painless way to get oriented to the world Stowe and her middle class readers shared is to look at the illustrations of that world, as they saw it, at least, if not how it actually looked, in popular publications. On this page, look at the illustrations for Godey's Lady's Book and The Mother & the Child At Home. In connection with the theme of temperance (below), you can also look at The Bottle, though these pictures were originally drawn in England.
As long as you're on a computer with a fast internet connection, listen to the first four songs on this page — though you might not call this way of gaining access to the period painless.
By far the most popular reform movement of the 1830s and 40s was not Abolitionism, but Temperance, and by the time Stowe began writing Uncle Tom's Cabin hundreds of temperance texts had appeared (including Walter Whitman's first book, a novel called Franklin Evans). Arthur's story and the following play were among the most widely known of them. (Arthur also wrote Ten Nights in a Bar-Room, which became the period's most frequently performed temperance drama when it appeared in 1854, two years after Stowe's novel.)
Read Act I, Scene 1, and all of Acts III, IV and V.
We should read more "domestic" texts, to see how gender was constructed in mid 19th century America, but as a representative text, read the first of the two stories on this page, "The Better Way." Mrs. Southworth was the most widely read American writer of either sex during the 1850s, and well beyond. She's best known in our time for The Hidden Hand, but this tale is more characteristic of her work.
If you want to see more on Stowe's middle class culture, help yourself.