In QFT, the Lagrangian is usually the form that's most useful, as this is what you use to calculate scattering amplitudes for processes. The Feynman rules for scattering processes come from the path integral formulation, which uses the "action", a quantity that's the integral of the Lagrangian.

My context is I'm (slowly) writing a quantum field simulator as a hobby. I've done this before for EM fields only, and am familiar with how to directly apply Maxwell's equations as a simulation. But the Lagrangian I have no clue how to directly utilize in a simulation. Hence my search for field evolution equations.

The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-... for example. Youll need to read a bit on calculus of variations. Usually in university this is gently introduced through the lagrangian formalism of classical mechanics

Ah thank you!! I've been searching for a straightforward derivation like that for a while. Maxwell's equations are familiar ground for me, so that is very helpful.

Well so this is for classical fields, where the standard model is a quantum field theory. So it will effectively follow the Schrodinger equation with the corresponding Hamiltonian from this Lagrangian. There are a lot of complications though, so I don't believe you will be able to just plug the Standard Model Hamiltonian in like this. Renormalization complicates things. You can do numerical simulations for certain parts of the Standard Model, specifically quantum chromodynamics (QCD), using lattice QCD, but that's not something I have experience with, so I can't speak to how easy that is to implement.