What were some of the challenges that women faced during the Women's Rights Movement?

There have been several waves of women's rights movements, including the wave in the early years of the twentieth century, when woman were primarily campaigning for the right to vote, and one in the 1960s-70s, in which women were primarily campaigning for equal career and educational opportunities. In both cases, a chief challenge women faced was engrained perceptions of "femininity."
In the suffragist movement, aimed at getting the vote, suffragists had to field complaints brought by both men and women that they were too militant, too "unwomanly," too "unnatural," and too willing to upset social norms. People argued that the man's vote should represent the entire family and that the woman should stay in her "sphere." These included even woman like Laura Ingalls Wilder, who would later become a prominent author.
The same problems of perception dogged the later women's movement. In this case, the real issues being publicized by activists—such as access to top rank colleges, i.e. Yale and Princeton, which were closed to women, and access to top career positions, which were traditionally "male only," as well as issues of equal pay for equal work—were sidelined, and the activist women were trivialized as "bra burners" and "man-haters." Women had to explain why it wouldn't be a "waste" to attend top colleges and aspire to top careers when their supposed "destiny" was marriage and motherhood. Some people, both men and women (such as Phyllis Schlafly), were appalled that women would want to "compete" with men.