Qualifications
Get ready to be certified. You may have steady hands (you never caused Cavity Sam's nose to light up), and super stress-management and people skills, but without the proper certification, you'll never put a surgical clamp on anything. Completing four years of college with a bachelor's degree in science is a good thing, but what most employers of surgical assistants are looking for is certification.
There are two options to get that certification: first, become certified as a surgical technologist (a lower rank than surgical assistant) and get three years of experience under your belt; or, complete a surgical assistant program. Both surgical technology and surgical assistant programs take one or two years to complete.
Once you have one of those certifications, you need to take a credentialing test offered by either the National Board for Surgical Technology and Surgical Assisting, or the National Surgical Assistant Association. To be eligible to test with these organizations, you must have completed your training as surgical technologist or surgical assistant through a program that is accredited by the Commission on Accreditation of Allied Health Education Programs.
Aside from all the programs and credentialing, really good surgical assistants have a bunch of other skills in their toolkits. To really excel at this job, you'll need steady hands and sturdy legs and the ability to work as part of a team (read: do every single thing the surgeon says). It wouldn't hurt to have a CPR certification as well.