Each hospital will be different. in most cases you'll be the nurses b!tch. Run here, get that, take so and so there. Some hospitals will allow you to do phlebotomy. Most will add EKG tech to your list of duties. I worked at a level 2 trauma center. Did more that my fair share of EKG's, blood draws, and vitals. It was great work experience in that I got tons of patient contact and used this to my advantage. The tech is usually the first person a patient will see once they make it back to the room. Use this opportunity to brush up on your interview/assessment/people skills. It'll also be a great opportunity to get a lot of codes/traumas/SHTF situations under your belt.
The one exception to working under doctors or nurses orders is if you are on a trauma team, in which case you may get to work under standing orders, starting an IV or doing bleeding control on patients. Its different from EMS, but should be less stressful and will give you some experience that can be applied to the EMS field.