I will add to the awesome answer provided previously. Other types of classes that I took in engineering school included programming. In today's terms, these would the types of programs used to control electrical circuits or physical movement of objects. My engineering degree is not in mechanical or electrical engineering...the programming courses I took were a part of the standard curriculum that other students in other engineering disciplines took.
Lots of calculus...surprisingly not as much physics as I feared :).
Hope you find this answer helpful. Best of luck to you!
Thank you for the question... To answer below are a couple core classes that are needed. One of my favorite was the Robotics class I got into the Lego Robotics and loved it ever since. I see that you are from St Louis. If you are looking to stick around that area I would recommend Missouri S&T for a Mechanical Engineering (ME) degree...789 of the 47,222 Mechanical Engineering diplomas awarded last year were given by schools in Missouri. It's located in a small town called Rolla.
Thermodynamics and heat transfer.
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Most colleges and universities have a rather set curriculum for engineering. This would involve math, chemistry, physics/engineering classes - statics & dynamics, electricity and magnetism, applied mathematics and then your specialization courses based on what you wish to pursue. I took machine design courses that helped me understand how to design machines such as engines and compressors. I would reccommend visiting their website to collect more information.