Priors are your previous knowledge on the topic.
> One textbook example of Bayes theorem is how doctors overestimate the probability of being positive for a disease. But what are the priors?
In this example, doctors overestimate precisely because they don't take the priors into account.
Doing something risky the day before / feeling funny is extra evidence that is assimilated (or should be) into the likelihood ratio P(D|H) / P(D). This is information the patient should share with the doctor.
Of course, if they don't, then the Bayes estimate is the best guess given all the information the doctor has.
Edit: Your criticism about how we choose priors is fair. The better you are at this, the more accurate your answers become. I mention more about this in the "putting it to practice" section.