What would you think about that?
This is a good question. I've spent a good amount of time during my Ph.D in Biostatistics consulting for academic physicians and their research. If you (and moderators) will allow for an opinion based answer then I'm happy to give it.
Medicine for some reason has created a culture in which the physician is intended to do everything themselves. Study design, data collection, analysis, writing, oh yea and on top of that their clinical duties and learning more about their specialty. These include responsibilities of an epidemiologist, data architect, statistician, just to name a few. Personally, I think that is a ridiculous onus to put on a researcher. This also might explain why medical research seems to be a copy-paste affair with bad statistics. Statistics is hard to learn, medicine is hard to learn, so learning both tends to mean taking shortcuts on one or the other or both (and understandably, it is the statistical rigour that is sacrificed).
Rather than succumb to these expectations it might be wiser to, as whuber notes, befriend a biostatsitician. Collaboration is a good way to learn, because you get consistent advice tailored to your specific situation as opposed to a mish mash of approaches from different courses with different learning goals. I'm not saying to defer all statistical work to a statistician, nor am I saying you should not learn about statistics independently, but I think rushing to learn all these things while also being a physician will lead to poorer work than if you were patient and collaborative.
The question is then "How do I meet/befriend a biostatsitician". Your medical school is likely attached to a university, in which there may or may not be an epidemiology department. Epidemiologists focus very carefully on how to do quality studies in a medical setting. THey should be well versed enough in statistics to help you out with design, data collection, and analysis. If you don't have an epidemiology department, there may be someone in a stats/math department, or in the sociology department (sociology is not exactly like biostatistics, but the difference between an epidemiologist and a sociologist grows smaller and smaller).
EdM makes a good point about the basis of fundamental probability and statistics. I'm not prepared to give a list of topics to learn and places to learn them. I think any undergraduate curriculum in science can give you enough to get started.
That being said, if pressed to offer one resource on a basis of prob and stats, I would recommend Introduction to Medical Statistics by Martin Bland. The book is geared towards medical students and in the introduction states
This book is intended as an introduction to some of the statistical ideas important to medicine. It does not tell you all you need to know to do medical research. Once you have understood the concepts discussed here, it is much easier to learn about the techniques of study design and statistical analysis required to answer any particular question.
The book however does not cover probability, and so you're free to pick up most introductory texts on the matter to cover that base. I agree with Bland that this book should serve as a good basis to read academic medical literature critically, and should serve as an excellent jumping off point to learn more about statistics in medicine.