You can start with

Hans Sterk
Finiteness results for algebraic K3 surfaces
Mathematische Zeitschrift, 1985, Volume 189, Issue 4, pp 507-513

The aim of this paper is to prove the following theorem:

Theorem. Let $X$ be an algebraic K3 surface over the complex numbers. Then
a) the group $\mathrm{Aut}(X)$ of (biholomorphic) automorphisms is finitely generated, and
b) for every even integer $d>-2$, the number of $\mathrm{Aut}(X)$-orbits in the collection of complete linear systems which contain an irreducible curve of selfintersection $d$ is finite.

If this does not give you enough, look at the references that seem promising and especially at the papers that refer to this one. Very likely anything newer that's relevant will refer to this.

Also look at the many papers of Nikulin on K3's. He has made lots of computations in the K3 lattice about automorphisms.

I believe Oguiso also has a paper where he studies finite automorphisms of Calabi-Yau's, but this is only a vague memory, so I could be off. Someone might correct me on this....

You can start with

Hans Sterk
Finiteness results for algebraic K3 surfaces
Mathematische Zeitschrift, 1985, Volume 189, Issue 4, pp 507-513

The aim of this paper is to prove the following theorem:

Theorem. Let $X$ be an algebraic K3 surface over the complex numbers. Then
a) the group $\mathrm{Aut}(X)$ of (biholomorphic) automorphisms is finitely generated, and
b) for every even integer $d\geq -2$, the number of $\mathrm{Aut}(X)$-orbits in the collection of complete linear systems which contain an irreducible curve of selfintersection $d$ is finite.

If this does not give you enough, look at the references that seem promising and especially at the papers that refer to this one. Very likely anything newer that's relevant will refer to this.

Also look at the many papers of Nikulin on K3's. He has made lots of computations in the K3 lattice about automorphisms.

I believe Oguiso also has a paper where he studies finite automorphisms of Calabi-Yau's, but this is only a vague memory, so I could be off. Someone might correct me on this....