Qualifications
You don't have to go to Med school at all to be a phlebotomist. You don't even have to go to college, if you pass the test and convince someone to hire you. However, you will have a much easier time at finding work with at least an AA degree and some certification or training.
Check the qualifications in your state. Some states, like California, require special certification and training.
Once you pass the phlebotomy test, you will be good to go in your job search. However, you will need to possess good coordination, a steady hand, a strong stomach, good people skills, a calm demeanor, a strong immune system, and the stamina to stand on your feet all day taking blood from people in a stressful environment.