I read a pop sci book on the Higgs which said that particles get their mass due to interacting with the Higgs field. If that is true, could we use first principles to derive the mass of, say, an electron? After all, QED is built on the interactions of particles and fields, right?
We could if we'd know the coupling between the Higgs field and other particles. Instead, we use the measured mass of particles to get the value of this coupling.
This is the table of the elementary particles of the standard model, SM.
The masses have been measured experimentally and the whole table is part of the postulates/axioms of the standard model. These masses within the SM are generated by the Higgs mechanism. The SM up to now is very successful in describing and predicting data. No theory has come up with a prediction of these postulated masses , while embedding the successes of the SM, so the answer is no for the elementary particles.
All other masses are composites of these elementary particles. For the mass of the hadrons the internal dynamics of QCD have to be used, and there has been a laborious lattice calculation that does give the hadron masses.
In the work presented here, a full calculation of the light hadron spectrum in QCD, only three input parameters are required: the light and strange quark masses and the coupling g .
At the moment there is no experimentally proven theory which could predict the masses of the elementary particles (for composed particles predictions can be made using mainly SM and to less extent the masses of the elementary particles) of the standard model(SM). The standard model does not make any prediction on the masses of its "members". That's one of the reasons why the physicists consider the standard model as not complete. One SM-extension (among a couple of other theories) is supersymmetry of which some of its variants together with the Grand Unified Theories (GUT) predict to some extent masses for the standard model particles (it's still not fully worked out), but actually there is no experimental evidence of supersymmetry (neither of GUT) at the moment. The particle masses still remain a mystery, in particular its bandwidth from neutrino masses of a few eV up to the top-quark mass of ~172GeV. The Higgs, so to say, only makes particles massive, but does not give us a clue about their values.