Geometric topology is not really a unique field in the way that algebraic topology and general topology are. Its various subareas may share something of a common feel (and indeed an arxiv category), but are often too diverse to have any common techniques. Those areas include, for instance:
Low-dimensional topology (classical knots, 3-manifolds, 4-manifolds, etc.)
Morse theory, simple homotopy theory and algebraic K-theory of spaces
Dimension theory (of separable metrizable spaces)
Topology of manifolds (surgery theory, codimension two knots, etc.)
Singularity theory (of smooth maps), geometric immersion theory (dealing e.g. with 4-tuple points of sphere eversions)
PL topology (block bundles, collapsing, bistellar moves, etc.)
Generalized manifolds, wild knots, etc.
Group actions on manifolds
Manifold structures (smoothing/trangulability and the Hauptvermutung; also Lipschitz structures, CD-manifolds, ... )
Embedding theory (smooth embeddings of projective spaces, PL embeddings of polyhedra,
etc.)
I surely forgot to mention many important subjects here; even the grouping of items in this list is rather arbitrary (and the order is random). The point is, you will probably not get far with diving in some depth into geometric topology unless you're more specific on what you're interested in. If unsure, try some knot theory or low-dimensional manifolds. These now cover more than half of all geometric topology by any count. Ryan and Jim gave some good suggestions of starting points in their math.SE answers, such as Rolfsen's 'Knots and links'. There are also other flavors of low-dimensional topology.
There exist some books and courses mentioning 'geometric topology' in the title, but they are often specialized and/or advanced. For instance, the 'geometric topology' notes by Sullivan and Lurie are mostly focused on manifold structures, and are firmly grounded in methods which are very clever and useful, but kind of external to geometric topology (localization, Galois theory and simplicial sets). Likewise, Bing's 'Geometric topology of 3-manifolds' and Moise's 'Geometric topology in dimension 2 and 3' are mostly about wild things. (There's definitely a trend in the literature that if geometric topology gets explicitly mentioned, things are likely not all smooth or PL.)
Arguably, closer to the point are Fenn's 'Techniques of geometric topology' and Ferry's 'Geometric topology notes'. Even these two virtually don't overlap with each other, so they are certainly not equivalents of some canonical algebraic topology text such as Spanier's or Hatcher's. But perhaps closer to such an equivalent than anything else that I can think of.
