Very little is known in terms of regularity theory compared with the level it is known for area-minimizers. However there are recent breakthroughs in other senses i.e. existence and rectifiability.
See
De Philippis, De Rosa, Ghiraldin. "Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies." Communications on Pure and Applied Mathematics 71.6 (2018): 1123-1148.
De Lellis, De Rosa, Ghiraldin. "A direct approach to the anisotropic Plateau problem." Advances in Calculus of Variations (2016)
...And the other related papers by this circle of Italians.
A key issue is the lack of a monotonicity formula for varifolds stationary with respect to an anisotropic integrand. This is discussed in an old and somewhat hard to find paper of Allard, but Allard proved a result that suggests that perhaps it is not useful to look for monotonicity formulae for other functionals.
EDIT: I should clarify a bit: There are some older regularity results when the integrand is assumed to satisfy certain nice ellipticity hypotheses. e.g. one of the main papers is:
Schoen, Simon, Almgren. "Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals." Acta Mathematica 139.1 (1977): 217-265.
The characterization of ellipticity/convexity conditions for these functionals is a lore unto itself, but the situation in which most progress was made by papers like the one above is this: If your minimizer is a Lipschitz graph then you have a Lipschitz function that minimizes some functional. Look at the Euler--Lagrange equation of that functional. Is it a nice uniformly elliptic 2nd order elliptic equation? If yes, then some of the same techniques as for area minimizers will open up. But some of the recent theorems like I mention above have weaker hypotheses on the integrands. So, more general notions of anisotropic convex integrands end up corresponding to the theory of elliptic equations with $$ a_{ij}\xi^i\xi^j \geq 0 $$ type conditions, rather than $$ a_{ij}\xi^i\xi^j > \lambda |\xi|^2 \qquad \lambda > 0\ \text{fixed} $$ type conditions. This is a tricky area of PDE, let alone GMT.