You can get a start in biomedical research, working as a technician. It’s what I’m doing currently, and I’m planning to enter a PhD program in another year or two. That would give you a taste for the actual pace of science, though there’s little opportunity for much of a career in the long term unless you get some kind of graduate degree.
Many labs will hire junior technicians for a year or three, though they prefer those with a bit of research experience under their belt (usually from a summer during undergrad). The kind of work you’ll do in a position like that really varies. Some labs treat a technician like the lab bitch, where your job will be to clean glassware and mix up media. On the other hand, in some you’ll be doing the same sort of work that a first or second year grad student would do – running and designing experiments, though you’ll be working on someone else’s project. That’s what I’m fortunate enough to be doing.
Most technicians land somewhere in between. For example, two of the techs in my lab are basically doing the grunt work of a large scale screen, where they repeat the same kind of assay, with minor variations, hundreds or thousands of times. This is more monotonous, but not too bad for a short term position.
If you had a good relationship with any of your professors, they’re good to talk to to find labs that are hiring. A good recommendation from a colleague goes far. And if that doesn’t work, you can simply start applying for tech jobs at major research institutions (hospitals and universities will post these jobs on their hiring pages). It doesn’t hurt to try to directly contact researchers who do work that you’re interested in, particularly if you write a good cover letter telling why you’re interested and showing that you’ve done your research and read a few of their papers. IME, that’s a bit more fruitful than trying to run the HR gauntlet, where you’re just another resume on a humongous pile.
ETA: In the better labs, techs will be acknowledged for the work they do, with opportunities to present their work at conferences and even get credit on publications.
ETA2: On the more medical side of things, the job title you might want to look at is “Research coordinator”. These guys do the legwork for clinical trials: talking to patients about the trial, enrolling them, making sure the regulatory boxes are ticked, analyzing the resulting data, etcl.