For a "more classical" point of view:
"An introduction to Invariants and Moduly Moduli " by S. Mukai.
Fogarty J. "Invariant theory" (Benjamin, 1969)
For a introduction to Mumford's:
P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces"
Anyway you have to learn (before or after) a Gothendieck-categorical background:
Fondements de la géométrie algébrique (Grothendick). The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)
Or in more gentle way: Fundamental Algebraic Geometry. Grothendieck's FGA Explained - Fantechi B., Göttsche, L., Illusie L.