Medicine & Health Sciences

Get a BS in engineering. As long as the school's engineering degree is ABET accredited, it doesn't matter at all which school you choose. There is no advantage picking an Ivy League over a state school - save your money. In fact, your best bet is probably Texas A&M.
Don't waste your time with an MS in engineering. It is a fairly useless degree. They'd give you credit for a year of experience and a couple grand extra. Let the company that hires you to pay for your grad school.
If you want to be a civil servant, you will need to get a co-op position at NASA while you are in college. NASA has very few openings and almost all of them are reserved for co-ops.
If you can't get a coop position, apply for engineering positions with one of the contractors that support NASA. You'll be doing the same job, just with a different badge. Most of the flight controllers for the ISS aren't NASA employees, they actually work for United Space Alliance (USA). At almost every NASA centre, many of the people work for USA, Boeing, Lockheed, Raytheon, Hamilton, and other companies.
Now, if you really want to be a research scientist, I don't know much about their entry path.