I think many of the suggestions above are great, but being a CNA isn't the only way to get into nursing school. I don't have a whole lot of experience in health care, per se - I've never worked in the hospital environment. What I do have is training as an EMT (which I didn't use much because at the time I was in late stages of Lyme Disease), experience in an animal emergency clinic, and experience (one summer) as a home health aide for two young men with Muscular Dystrophy. At this point, my biggest background boost is that I work at the National Institutes of Health doing public health research funding.
And I'm about to start an accelerated second bachelor's program at Georgetown.
So, if you don't want to do the CNA deal, there are other options. I'd suggest some home health aide work (keep in mind pay is low, but it can be great experience). You might also volunteer at the Red Cross. Most volunteer work in the hospital setting that I've found isn't patient-care related at all.
I do feel strongly that you should get some sense of patient care before you actually apply and get to school. As harsh as the previous poster comes across, it does make sense to know what you're getting in to. Don't just think that nursing, no matter what level, is all butterflies and rainbows. It's a tough, sometimes dirty, "roll up your sleeves and push back your hair" kind of job.
Take some time to figure out if you know why you want to apply. That's usually a huge component of being accepted (in my experience).