How did one become a doctor prior to the established medical system circa 1910? i.e. How did you become a doctor in 15th century etc?

I cannot speak for the fifteenth century, but I can speak for earlier medieval eras!

Doctors were doctors because people came to them for medical needs. They didn't need to study anything, necessarily. If they were able to help people get better without a day of study in their life, people didn't care. They just didn't want to be sick. However, doctors wouldn't get much business if they couldn't treat people, so usually they studied out of necessity. There were medical guidebooks in existence, which included knowledge of anatomy (as they believed it), pharmaceuticals (as they believed it), and diagnoses (as they believed it). Some doctors studied from these books, and added upon them with their own experience from what they learned through practice. Some doctors apprenticed with another doctor to learn their craft in that manner. Some learned it from their parents (women as practitioners of medicine is definitely not unheard of).

As for what they healed, it was a range of things just like modern doctors deal with. There are cases of mental illness, cancer, strokes, various skin disorders, STDs (sexual health is my favorite medieval medicine), stomach illnesses, wounds, everything you can think of. What did they heal effectively? That's far more questionable. They were usually pretty good with things we don't need to see a doctor for in modern days. Mundane rashes, stomach bugs, headaches, sleeplessness - they had figured out medicines that could help with those. But bigger things? Chronic things? It was more hit or miss as to whether the doctor actually did anything. Blood letting was hugely popular for centuries. Bathing was also a common remedy. Often just being in a clean environment, receiving adequate nutrition, and bathing did wonders for the average persons health, and that's something you could find at a classical or medieval doctor's house/office.