Not all private EMS is boring. I work for a private service attached to a local hospital and we do rescue stuff as well. Granted, we don't have turnout gear or cutters/spreaders, but we do SAR, HAZMAT decon, low angle/tough terrain rescue, ice rescue, tactical EMS support, and we have a dive team. On the whole, we can do more than the average volunteer fire dept. we work with, but we're somewhat unique in that sense.
Believe me, we all started out doing most of the driving. It's the bottom rung of EMS but as my boss put it, probably the most important. EMS's primary job is to get the patient to the hospital, which means good driving is critical. Get stuck in traffic or worse, crash the unit, and bad things may happen. Once your coworkers get more comfortable with you, they'll let you do more.
There are a number of ways to get to play more. First, join a volunteer fire dept. if you can. Many will take you off the street and train you. Some may even give you a small stipend for uniform costs and on-call time.
Next, get your I or P license. It's a lot of work, but you learn a lot and really get to do some neat things. Once you're a medic, you can join a professional fire dept. as a FF/Medic and really get to work some interesting calls.
You can also work in a hospital as an ER tech. You'll develop some great assesment skills very fast, which are at the heart of any good EMT's routine. You may also be able to start IVs, give nebs, etc.
Working in the ER as a tech, phlebotomist, whatever, is the best way to get accustomed to nursing. Patient assesments, vitals, and labs are at the heart of emergency medicine. Once you figure out what's wrong, the rest is pretty routine. Even (many) doctors have protocols and guidelines.
If you're really that disappointed with your job, talk to your supervisor. If he doesn't give you the right answers, start looking for a different service to run with.