Here is a version of the table of contents of Bowditch's lecture notes A course on geometric group theory.
Group presentations, free groups, abelianisation.
Cayley graphs.
Quasi-isometries and their invariants.
Fundamental groups, covering spaces.
Hyperbolic geometry, the plane, the space, surfaces, three-manifolds.
Hyperbolic spaces, trees, the four-point condition, exponential growth of distance, quasi-geodesics, Hausdorff distances, qi-invariance, hyperbolic groups and their properties.
Isoperimetric functions, linear bounds and hyperbolicity, qi-invariance, Dehn functions, the word problem.