Free groups and presentations. Primary source: notes by Inna Bumagin. These notes were in turn based on notes by Alexei Myasnikov.
Presentations of symmetric groups. Primary source: Johnson, Presentations of Groups (locked).
Knot groups: Wirtinger presentations. Primary source: my own notes, which were based on Hatcher's text, p.55 or thereabouts.
Quotients of knot groups. Primary source: Johnson, Homomorphs of knot groups (Proc. AMS 1980) (locked). Secondary sources: papers by Gonzalez-Acuna (locked), Neuwirth (locked).
Groups generated by reflections. Primary source: Coxeter and Moser, Generators and Relations for Discrete Groups (locked).
Changing presentations without changing the group (Tietze transformations, Andrews-Curtis moves). Primary sources: the Tietze transformations page at wikipedia.org; Rolfsen, The Poincare conjecture and its cousins.
New groups from old: free products, amalgamations, graphs of groups, Bass-Serre theory. Primary source: Scott and Wall, Topological methods in group theory (locked).
The Ping-Pong Lemma, faithful representations of free groups. Primary sources: notes by Marc Culler on free groups (public), Macbeath, Packings, free products, and residually finite groups (locked).
Valuations, the tree for SL_2(K), K a field with discrete valuation. Primary source: Shalen, Representations of 3-manifold groups, preprint version (public), final version (locked).
The tree for SL_2(Z). Primary source: Dicks and Dunwoody, Groups acting on graphs (locked).
Linear representations of finitely presented groups. Primary source: Shalen, Representations of 3-manifold groups.
3-manifold groups acting on trees and incompressible surfaces. Primary sources: Stallings, Group theory and 3-manifolds (locked); Shalen, Representations of 3-manifold groups.
Hyperbolic 3-manifolds. Primary source: Thurston's Princeton notes (public), my own scanned, marked up copy (locked).
Incompressible surfaces and boundary slopes. Primary source: haven't found a good reference yet. Secondary source: Hatcher, Boundary slopes of incompressible surfaces (locked), updated version (public).
The A-polynomial. Primary sources: the two papers listed below. For an interpretation from the point of view of physics, look at a paper by Gukov (locked).
The papers/topics motivating all of this:
Culler and Shalen, Varieties of Group Representations and Splittings of 3-manifolds (locked).
Cooper, Culler, Gillet, Long, and Shalen, Plane curves associated to character varieties of 3-manifolds (locked).
Cooper and Long, Representation Theory and the A-polynomial of a knot (locked).
Shalen, Representations of 3-manifold groups, chapter 19 of "Handbook of Geometric Topology (locked).