What Lizzy says is a good starting point!
In short - there are research opportunities throughout medical training and in most settings. If you have the "fire in your belly" you can do as much or as little as you want from going MD/PhD and working the classic "80/20" - 80% research with 20% clinical duties to having a straight MD education and full-time clinical privileges with research being "your hobby" nights and weekends (not to belittle their work, but you get the point). Regardless, it should be pointed out that research invariably requires more training - a year off for research, other degrees, summers of training, a research-track residency, research fellowships, etc. etc. It doesn't substitute for medical education, but augments it.
There are examples of very permutation in between of successful researchers. You need to choose a lifestyle and career path that achieves the balance you are looking for (and this may vary over your career and education).
As for "where to start?"
1) first think about what you want your career to look like - predominantly research or clinical? Academic (where you will teach) or not? Private practice?
2) Then figure out the types of research interest you: basic vs. clinical vs translational. Note that there are other areas ripe for research (Lizzy alluded to these with the "JDs and MPHs") - ethics and law of medicine, the economics of healthcare and healthcare delivery, Public policy.
3) Finally, look for good mentors who have achieved/developed a career that seems appealing to you (a good combination of these factors above) and ask for their guidance. Awesome researchers don't just happen - they have had and continue to have excellent mentoring throughout their careers (life-long learning).
Mind you, you might have to take some "lumps with your lessons" - getting good mentors and breaking into medical research can be difficult. But research can be extremely rewarding and can enrich your career in medicine - and vice versa!
Best of luck!