Qualifications
There are no undergraduate degrees for geothermal engineering (source). So, great, no degree required, right? Wrong. You're going to need schooling, and lots of it.
For starters, you need to not only take math and physics in high school—you need to excel at them. Then get ready for the chock-full-of-fun world of engineering school. If you really want a diploma with the word "geothermal" on it, you can attend a graduate program like Stanford's in the Department of Energy Resources.
Once you've sacrificed the majority of your time, energy, and soul to your education (not to mention the entirety of your bank account) and have found a job, you have every right to call yourself a geothermal engineer. Well, almost every right. Do you want to be a licensed geothermal engineer? Then wait, there's more.
To become a Professional Engineer (those are the industry's capital letters, not ours), you need to take and pass the Fundamentals of Engineering exam after graduating, then notch four years of work experience, and then take a second exam on the Principles of Engineering. See? Piece of cake.