Pre-requisites and references for $K3$ surfaces

K3-surfaces are non-trivial examples of compact complex surfaces, i.e. manifolds of complex dimension 2. Hence you need some background in the theory of several complex variables or in algebraic geometry. The first topic is the subject of

"Barth, W.; Hulek, K.; Peters, Ch.; van de Ven, A.: Compact complex surfaces."

The second theme is the subject of the classic

"Hartshorne, R.: Algebraic geometry."

For both themes you need sheaf theory and cohomology theory as a prerequisite. You find them in Hartshorne's book. In addition, some background from Riemannian surfaces, i.e. manifolds of complex dimension 1, and curves, i.e. complex spaces of complex dimension 1, is helpful.

If you set up to master K3-surfaces, you will find on your way many examples from surface theory like tori, ruled surfaces, elliptic fibrations or Hopf surfaces. These examples are easier to handle and are interesting by themselves.

In the end, you find the theory of K3-surfaces in Chapter VIII of the above mentioned book on compact complex surfaces.

Here is a great reference I found (in case someone might be interested):

http://www-math.sp2mi.univ-poitiers.fr/~sarti/corso_Perego.pdf