You need research experience from home to get in with these programs. I am working with a prestigious summer research program and I wouldn't have had it if I didn't have research experience back home. Start looking for opportunities in your school first. Apply your junior year for the summer research programs. They prefer seniors (rarely juniors) because by that time you have enough scientific knowledge to tackle complicated projects.
That preference is actually changing at many schools because it's too late (by the time you're between junior and senior year) to really sway you into choosing their school/program, particularly for programs that get a lot of MD/PhD hopefuls (for PhDs, they still have time).
The Mayo Clinic's program had about 1/5 -1/4 rising juniors when I was there and I was told by a program director that they were increasing that amount. As such, much research experience is not necessarily required. Case in point, there were maybe 5-10 out of the 100 or so at Mayo that had zero biomedical research experience (though I'm sure they had killer recommendations from lab instructors or something).
Based on the two "prestigious" summer research programs I've been a part of, these programs, numbers-wise, are as competitive as the MD/PhD admissions (a few percent accepted). However, there is also no charge for applying and no one seems to keep track of how many people applying to formal programs actually get in somewhere (I imagine it's better than the MD/PhD application cycles).
Things that make it easier to get into programs:
- Look up the mission of the graduate school/med school/etc. and tailor your application to it
I applied to both as a pre-MD/PhD and did best at programs with a strong clinical research component to their missions. Make sure you know what you're getting into (the program itself as opposed to any of the other prestigious programs) and make sure they know that you know.
- Apply locally/regionally
I had a much higher acceptance rate in my region. Just as with any other admissions, they want to know you'd seriously consider accepting if they offered you admission. The admissions happen rapid-fire (as in you have a week or two max to decide if you want to attend once they offer you a spot), and they want to catch the best applications before they're taken by another school so it's even more important than applications to grad/med school that you're a good fit and seem willing to go. That said, I'm at a program on the East Coast that has 10% Midwestern students and 5% West Coast students, so it's possible to get in as a "diversity student."
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- Apply to programs with smaller stipends and especially those with no room/board provided
Great applicants will apply broadly and, all else being equal, will choose programs that pay the best/create the least inconvenience (i.e. finding a sublease on Craigslist or calling every apartment in the area looking for a short-term lease). If you're willing to break even or straight-up invest in a summer at a great research program that doesn't pay as well (you mentioned volunteering), then you'll be able to score great experiences even if you don't make any money.
- Do you research on StudentDoc
There are a number of posts with people talking about which programs they got into (acceptances threads). Send them a message asking about the composition of the program (Did most have a prev research experience? How many were PhD vs. MD/PhD vs. MD? Were they mostly regional or from all over?).