It depends on what industry you are interested in. The qualifications are different if you plan on a career in architectural, automotive, audio equipment, environmental, underwater, or aerospace acoustics.
Some of the fundamental qualifications include understanding the mechanics (physics) of sound propagation, understanding the relationship among acoustical terms, including SPL, dB, near field, far field, etc., understanding how to use a sound pressure level meter, microphones, and sound intensity measurement equipment, and understanding of some of the analytical techniques to predict sound propagation, including statistical energy analysis, boundary element, finite element, etc.
Once you decide on what branch of acoustics that you want to pursue, it wouldn't hurt to contact some of the leading companies in the field and ask them for their opinions on what sort of background you need to enter into the profession. You would be surprised at how cooperative some of these companies may be.
pj