To "understand cancer on some very serious advanced level" is a major endeavor for someone coming from outside of a biological sciences background. You might look at it that biology and chemistry are the foundation of medical sciences, where physics and math are the foundation of engineering sciences. Physics and math are of course important in medical sciences, but I might say that it's only insofar as their role in biology and chemistry.
Biomedical engineering is where the two really start to cross over. If you are considering focusing on cancer with your career, you might consider studying Biomedical engineering - but either way, I recommend you take a look at the coursework of a Biomedical engineering degree to see what you HAVE and HAVE NOT studied yourself.
If I had to break cancer into fundamentals: when a malfunction occurs in the genes (or "processing" of genes) in the cells of a tissue of an organ in an organ system of an organism, each specific malfunction can cause specific normal histological structures and physiological functions to go "wrong" in specific and complex ways that have specific and complex effects. There are specific conditions that can cause those genetic malfunctions to occur. And there are a LOT of things we still don't understand about many of those steps.
Therefore, to truly understand cancer, you need to understand many layers of material that are rooted in those below.
- Pathology requires knowledge of histology and anatomy.
- Pathophysiology requires knowledge of physiology - not just of the organ/system in question, but also general endocrinology (signaling pathways).
- Physiology and histology both require cell biology, molecular biology, and genetics.
- All of those require a solid grounding in biology and biochemistry.
As has been suggested, it may be easier to start with a specific cancer and start making a list of concepts you need to read about when you find your gaps. OR you might look at TED talks or university lectures posted online for a video on the fundamentals of cancer, and see what you understand already vs what you would need to learn.
I can only offer a single person's opinion, so take it with a grain of salt, and I am light years from being an expert in cancer. But the first time I really felt that all the concepts fully came together and sprung alive was a lecture in a pathology review course called Pathoma - and that was my second year of med school after an undergrad degree in biology. That's also when I grasped that there is SO much more to it that I will never understand without dedicating enormous time to it. He might offer a free video, I don't know, but it was the best framework I'd seen.