Personally, I have a degree in theater tech/design. So, it is fairly related to stage management. However, many of the people I know who are professional stage managers do not have degrees in theater. Many of them were English majors or something else in college and worked shows in the theater. One of my colleagues has a degree in mechanical engineering and another in psychology. So, stage managers come from all areas. Like most of the people here have said, find some theaters that will hire you as an intern or production assistant and start to build your resume that way.
I am primarily of the no grad school position, but if there are skills that you feel would be worthwhile having some schooling on then it is something to look into pursuing after a year or so out in the world. Since you did not have the college stage management experience grad school would also be a good safe zone for honing some of the stage management instinct as well.
Whatever you choose to do best of luck!