Likely this should only be a comment, but I don't have enough reputation for that...
J.C. Ottem has provided a wonderful reference about the basics of K3 surfaces in his comment. It's my personal experience though that when I'm working through notes such as Huybrecht's, it's instructive and motivating to have short and interesting papers to read once I've learned enough theory. If this is also the case for you, I recommend having a look at Deligne's ``Relèvement des surfaces K3 en caractéristique nulle''. The article clearly explains the degeneration of the Hodge to de Rham spectral sequence and is a nice way to start using some characteristic $p$ methods and deformation theory.
I should also mention that there are some cool notes here:
and point out that there's a list of references at the bottom of the page which you might like to browse. The notes also include a discussion of some uses of K3 surfaces in physics, if that interests you.
One of the Arizona Winter School 2015 courses was ``Arithmetic of K3 surfaces'', given by Várilly-Alvarado. Not only are the notes nice, but there are also videos of the lectures. Here is the link...