What general courses would a successful mechanical engineer take in college?

Differential Equations, Calculus (and Multi-variable Calculus / Linear Algebra), Physics (Mechanics and E&M - electricity and magnetism), Thermodynamics are the main ones I can think of, off the top of my head.

The career paths you have afterward are not indicative of these general course requirements to major in Mech E in college.

I was an engineering major in college.

Mechanical engineers need to know about a lot of different topics, including architecture, design, CAD, physics, and material science. But underlying all of these fields is a basic need for mathematics. As your career progresses in mechanical engineering, you’ll notice that the people that really get ahead and do well, and get paid very well, typically have strong knowledge of calculus, and differential equations. Why? Because these types of mathematics are very useful in modeling moving parts in complicated machines. If you want to build engines, robots, or high tech machines, you’d better take a differential equations course at some point in college – you’ll thank yourself years after you graduate!

A successful mechanical engineer will take the basic STEM-oriented undergrad courses required by that major. That is, a sequence of mathematics to higher mathematics, statics and dynamics (the study of systems and mechanical architecture), some chemistry, and mechanical engineering itself. That being said, it's a good idea, IMO, to take some electronics, physics and computer science. There are very few jobs where computers will not be used, and anyone who goes through an engineering curriculum without getting fairly deep into the computers is cheating themselves of opportunities.

Now, one of the complaints I've heard about engineers (generally from non-engineers) is that they are science and fact oriented and not very good communicators. While my own experience has been that they are indeed good communicators, I'd say there's a very good reason why successful ones are. That's because technology, many other human efforts, have consistently raised the bar. That is, projects get bigger, more ambitious, more complicated. And the numbers of people required for any particular project get bigger. And the interfaces between people on these teams or projects increase in number geometrically. People have to be more efficient communicators. They need to be good speakers, good writers, clear explainers, people-oriented and efficient in documenting what they do and did. Otherwise, we'd each enter each new task from scratch without knowing what the people before us have already found out.

You'll see that the most successful engineers, in general, are good communicators. So make sure you take care of that and get comfortable with it. It helps every team be better.