Mass, size, temperature, luminosity, chemical composition, the initial abundance of the molecular cloud, distance, brightness, age, and evolutionary cycle can all be used to characterize a star. A star's brightness is dependent upon its distance and luminosity, so brightness is not a "first principle"; however, there are some properties that seem less obvious in determining their dependencies. For example, chemical composition and temperature seem to be at a constant interplay. Does one always inform the other? Chemical composition and mass influence a star's evolutionary cycle, so it's evolution through the HR diagram is also not a first principle (from my understanding).
For fun, if we wanted to describe a star in the most simplistic way possible, which properties would we need to know? My assumption would be the initial chemical composition when the star was made and its initial mass. If we wanted to know the star's brightness for observational purposes we would also need to know the distance, but this isn't a fundamental property of the star. In other words, what are the basic properties we need to know to extrapolate the rest? I'm, betting on mass and initial chemical composition, but am unsure about temperature. We can throw in distance for observing properties but I was mainly using this as an example.