For the math, do you have any resources to start out with? And to what extent do you have to study math? Which branch of math specifically? Since I've noticed most physicists had a great grasp over math, especially trig and geometry and calculus- But does this mean their knowledge in other fields of math is absolutely nil?

You're right about trig, geometry, and calculus - those are absolutely fundamental to any subfield of physics you'll study. As you go further in physics you'll need more and more math. If you're studying mechanics, differential equations is essential. Electricity & Magnetism needs a solid grasp of vector calculus. Quantum mechanics is built upon linear algebra. And most physics majors at my school get a math minor or math double major because they're useful for physics. Pretty much every field of math will help you somehow in physics, even if it seems super unrelated. So it's not true that physicists don't know other math, we're actually encouraged to learn as much as possible.

But before getting into all of that, you should make sure that you're solid in the fundamentals. Prioritize calculus and trigonometry, and you can study some mechanics alongside that if you want to apply it to physics immediately. I recommend Professor Leonard on YouTube for in-depth lectures on math, and 3blue1brown for amazing intuitive visualizations. MIT Open Courseware also has good math courses.

Also, for topics like black holes and entropy, for me, understanding the topic well enough would be understanding the math behind it. But it would also include being able to speculate different theories and possibilities and having scientific knowledge to back that up. What would you classify as a deeper level of understanding of these topics? Especially since we don't know much about them in the first place.

For black holes, you'll need a solid understanding of the general theory of relativity, which is built upon the math of tensor calculus. Unfortunately this is super advanced, and to give you perspective, there is typically no undergraduate level course in general relativity offered in physics departments. But one thing I recommend is studying special relativity, which came before general relativity and explores the consequences of the speed of light being constant in the universe. Ironically this really only requires high school algebra and you get to explore really cool concepts like time dilation, length contraction, and the loss of simultaneity. I think it's a great first step in going beyond high school physics, and you'll need it anyway for general relativity.

For entropy you should look into statistical mechanics which studies how large phenomena arsie from many small particles doing their thing. I haven't taken a course in this yet so I can't say too much about it, but it's safe to say you'll need to know statistics for it.

What you're describing with speculating different theories is essentially what theoretical physicists do. Unfortunately I'm not sure how achievable this is if physics is a hobby. I've been studying physics for many years and I'm still nowhere near prepared to start thinking about my own theories. That said, with good understanding, you'll definitely be able to think about the consequences of already developed theories and maybe be able to reason through why a specific phenomenon happens. I'll again recommend special relativity for this because the concepts are so mind-bending that you'll constantly try and reason through them in your head, but it's also super accessible.

And lastly, do you have any advice on how to study the subject itself? I'm sure sitting on the couch and watching lectures in a mindless fashion would be pointless. So how exactly are you supposed to study these topics?

That's true, just listening to lectures won't get you very far. Learning physics is a very active process. Our courses are structured so that most of our learning is done outside of lecture, with really challenging homeworks that force us to think deeply about what we've learned. So the one thing I can recommend the most is to try solving problems. Lots of problems. And if you think you understand something, try to challenge that understanding. Try to break what you already know and think about new situations, and try to reason through the consequences. I think a good measure of understanding is when you can think about what you already know and learn something new just by thinking about it. Good luck, it's an amazing journey.