Below is a set of notes on multiple classical and modern references on Gromov-Witten Theory and Quantum Cohomology. Here is a basis of ideas which play an important role in mirror symmetry.
"Introduction to Quantum Cohomology" by Joachim Kock and Israel Vainsencher
Notes
Chapter 0: Prologue: Warming up with Cross Ratios and the Definition of Moduli Space
In this chapter we introduce the first nontrivial example of a moduli space M_{0,4}, the moduli space of quadruples of points up to rational equivalence. In particular, we introduce notions of families and pullbacks and generalize the moduli problem to study fine and coarse moduli spaces. Here we introduce the original motivations of classification problems in algebraic geometry from the categorical perspective as founded by Grothendiek, Knusden, Degline, Mumford, and many others! This story, although not comprehensive, gives a first look at how these amazing mathematicians formalized using geometry to study classification problems.
Chapter 1: Stable n-pointed Curves
In this chapter we introduce the natural compactification of the fine moduli spaces M_{0,n}, the moduli of stable curves with n marked points. This notion was introduced in the algebraic geometry setting by Kontsevich. Stable curves of n marks form a fine moduli space with rich recursive structure on the boundary. One main reason we enjoy these properties of a compact moduli space is for the purpose of doing intersection theory of curves in the Moduli. The key insight here, known of course by Grothendiek, is that the addition of data on your algebro-geometric objects in your moduli problem kills all nontrivial automorphisms of any particular object -- giving each object a unique "name"/"identity"! This of course is extremely relevant in practical uses of moduli spaces, especially enumerative applications as you'd like a good understanding of what you are counting and how you are counting it. If an object has multiple automorphisms, how do you count it? With multiplicity? As a single object? With stability conditions we don't need to worry about this phenomenon.
In light of the previous chapter, the goal here is now to generalize the notion of stable curves as embedded in projective varieties rather than just sitting in an arbitrary ambient space. Here we get the additional data from the embedding map, in particular, it's degree on each branch of the domain tree of rational curves. Again, we introduce a stability condition to compactify the coarse moduli space of such objects. It is known that there is a much easier presentation of this moduli space as an algebraic stack, however, for our purposes neither explicit construction is necessary. However, with a good understanding of how we can appropriately obtain stable embeddings of trees of rational curves of certain degree we are now able to count curves in P^2!
Chapter 3: Enumerative Geometry via Stable Maps
Chapter 4: Gromov-Witten Invariants