I think you need about 3 courses and maybe some application. Then it helps and you end up not believing anything you hear because you realize how contextual and subjective things are, and that is excluding the myriad of biases.
I think statistics was the best thing I ever learned. Some benefits: able to spot lies, hype and plain BS in many different fields. Great framework to find the right questions to ask. Steady intellectual training, to always check the data before giving in to emotions...and so on.
There is a famous Wells quote from the 1910-s saying that in the future the knowledge of statistics will be just as indispensable as the knowledge of reading and writing. Apparently it is a fake quote but still true.
I hardly know anything about statistics compared to the regular posters here. But I would say that my limited knowledge of probability informs my thinking about everyday life things, whereas my limited knowledge of statistics does so very rarely.
Partly that's because I see probability as answering questions that lots of people naturally have about their world (i.e., what are the odds that X will happen to me if I make decision Y, that my kid will inherit trait Z, etc.), whereas statistics seems to usually make people aware of a problem they wouldn't have realized on their own and then presents a solution to that (i.e., you thought doing your experiment was as simple as comparing the control group and experimental group means, but WAIT! ACTUALLY ANY DIFFERENCE YOU OBSERVE MIGHT BE PURE SAMPLING ERROR! We'll now proceed to talk at length about strategies for getting around this problem).
I always thought that "probability" was interchangeable with "statistics". Example: The math department labels the first course "Probability" while the stat department labels the first course "Stat 1". Only difference between the two, the math class was more theoretical.