Which stellar properties can we describe as "first principles" in which we can derive the rest?

Rob Jeffries has covered pretty much everything in his answer, but I'll add that this is a question with a long history, enough that there is a famous answer to it called the Russell-Vogt or Vogt-Russell Theorem. That states that composition and mass are the two key properties, assuming that the star is in hydrostatic equilibrium and derives its energy from nuclear reactions.

I agree with Rob that rotation (angular momentum) is a key second-order effect, and I'd add magnetic field strength as well.

The Vogt-Russell theorem isn't really a theorem in a mathematical sense - see here and here for a deep dive on that and the history of this question - but it captures the fundamental simplicity of stars (even though at the same time they have a lot of interesting complexity in the details).

Starting from a protostar, one would hope to be able to predict everything about its future development if we knew its initial mass, chemical composition and angular momentum.

Mass is fundamental because it determines how much fuel the star will have and the pressure at its core. Composition is key because among other things it determines the opacity of the stellar material. Angular momentum is important because rotation can alter the structure of the star, if fast enough, and can also enhance internal chemical mixing.

Of course extrinsic factors like interactions are also possible, so the properties of any companions are also fundamental, as might be the birth environment that can do things like hasten the dispersal of circumstellar material from a protostar.

If you are talking about figuring out the past and future of a star observed midway through its life, then in addition to the above, we may need to know how the chemical composition varies with depth inside the star, since it will not be uniform.