Phlebotomy and nursing are two separate careers, but many of us have worked as phlebotomists on the way to becoming nurses. A phlebotomy course often takes no more than a few weeks at a community college, at which point you can take a national certification exam offered by any one of several laboratory agencies. (See
www.ascp.org for one such agency.)
As a nurse, your pay and job opportunities will be much better, and nurses still do the blood draws in many settings. There is even a specialty of nursing (IV nursing or infusion nursing) which deals with placing IV lines. IV nurses often get called in on patients who have tricky veins, sometimes after multiple attempts have already been made. There's also a specialty within nursing known as apheresis (or just pheresis), where a machine is used to slowly withdraw blood from a patient, spin out a certain component of the blood, then return the rest of the blood to the patient.
In short, there are many opportunities for nurses to perform phlebotomy-related tasks like drawing blood.