I would like to know what can be said about the classification of torsion-free modules.

For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does one proceed when trying to classify finitely generated torsion-free modules over $R$, or equivalently torsion-free sheaves on $\operatorname{Spec} R$? What general theorems are available, do there exist moduli spaces and what do they look like?

I have seen that for $R=k[X_1,X_2]$, there is some relation between torsion-free sheaves on $\mathbb{A}^2$ and the Hilbert scheme of points on $\mathbb{A}^2$? Unfortunately, I have not found a precise statement. Where is the relation between isomorphism classes of torsion-free sheaves and points on the Hilbert scheme of points made explicit, and proved? Are there some classification results for torsion-free sheaves that generalize to affine spaces of arbitrary dimension, or even to smooth affine schemes in general?

Finally, what can be said about automorphism groups of finitely generated torsion-free $R$-modules, automorphisms taken of course as $R$-modules? Are there any structural statements, either from the algebraic group or the discrete group point of view? Again, I would be interested both in the special case $k[X,Y]$ as well as in the case of smooth affine schemes.

[Edit: Thanks for the answer and comments I received so far. I would still like to know more about the automorphism groups of torsion-free modules. Are there any statements about this in the literature?]