The general history of the women in the 1920s and 1930s, as summarized well by pohnpei397, probably applies well to large urban centers in the North but not so well to the more extensive rural areas in the South. There's no hint of the flapper in To Kill a Mockingbird, for example!
I think interviews and work in cultural studies are great resources for answering this sort of question.
The first link takes you to an interview of three women who grew up "white" in the South in the 1930s. The second link give information from black women's perspective. The third link leads to what looks like a student paper; a number of the sources used are online, so you may find some good information here as well.
There was not one single role for women in any time or place in history. Things differed to some extent based on the family of the individual woman, whether they were urban or rural, etc.
In general, women's lives in the 1930s were "opening up" a bit, especially early in the decade. The 1920s had seen a big cultural change that had helped women become more free. It had become more acceptable for women to do various things in public that they had not been allowed to do before.
Later in the 1930s, the Great Depression changed everyone's lives. Women had a much harder time then because they more often had to work (if they could find it) to allow their families just to survive.
So -- women were still not equal to men legally or in fact. But people's atttitudes towards women had loosened up some by the 1930s.