I will second what late in the game has posted here, as I completed a Master of Public Health degree a couple of years ago, and I'm attending medical school next year. Typically, MPH programs are not super competitive to get into, as I knew folks from a number of different kinds of institutions, with different backgrounds, and a variety of GPA and GRE scores there. However, some areas of concentration may be more competitive than others. But the graduate school process pales in comparison to the medical school process. I will refer you to the MPH lounge for more information on specific programs and schools.
I have considered pursuing the MBA as well, but I don't think that any Business degree would necessarily be a piece of cake, esp. if your background is not in business (my undergraduate was Biology/Sociology, and graduate concentration was Health behavior/Health education). Plus, the GMAT is an entirely different monster to tackle when applying to business school. Here, I will refer you to the MBA website, which has good general information about schools and the GMAT:
http://www.mba.com.
With this, if you are truly interested in health care administration, I might suggest pursuing a Master of Health Services Administration (MHSA) program at a school of public health. I will not advocate that it will be easier than pursuing a business degree (MBA), but it would afford you some of the advantages that late inthe game has proposed (although MHSA programs tend to be tightly packed and leave little room for additional coursework, etc.). If you do decide to go with the MHSA, the program at my alma mater, Michigan, is very strong. If you're not applying to med school immediately after graduate school, then you could probably get a pretty competitive administrative residency with a top notch health care system
😀 I have a few friends in these fellowships now and they get unbelievable clinical exposure since they generally have to do rotations at every unit within their health systems. May be worth looking into.
Best,
H&T