Part of the problem with geometric topology is that there still is no wide agreement as to what exactly the field consists of. Some aspects are parts of differential topology/geometry,some belong to algebraic geometry and others belong properly to algebraic topology. Frankly,I think the term "geometric topology" refers to an approach rather then an actual separate field. To me, a good working definition of geometric topology is that it refers to those aspects of topological spaces-particularly manifolds-that can be studied either by combinatorial methods,as in classical low dimensional topology, or by the direct generalization of these methods,such as piecewise linear structures and their characterization by their associated cell decompositions.
The best beginning source for a geometric approach to topology is the wonderful book Classical Topology And Combinatorial Group Theory by John Stillwell. In addition to the very modern course notes by Jacob Lurie recommended by user32240 above,I can heartily recommend the following links page, which contains a wealth of lecture notes and online texts on topology-including not only links to 2 courses taught by Lurie, but many other classic jewels in the field including hard to find lecture notes on differential topology dating to the 1960's by C.T.Wall, the classic text on PL topology by Hudson, complete sets of all the classical lecture notes by John Milnor at Princeton (!) ,Dennis Sullivan's classic 1973 MIT lectures and much much more. I think you'll find it immensely helpful.
Lastly, I'd like to add a recent discovery: John Francis at Northwestern University has posted a number of lecture notes on geometric and advanced algebraic topology at his personal page-they look quite good and are worth a look:
Clarification: Many of the sources above are quite advanced,but many-sucb as the notes by Zeeman and Wall-are not and should be accessible to beginners. The book by Hudson is the logical next step in the subject after Stillwell and a course in differential topology.The notes by Francis are fairly advanced and require at least a semester of a graduate course in algebraic topology a la Hatcher for their full comprehension,but there's a wealth of material on the applications of the Whitney trick to low-dimensional manifolds there,which the OP specifically asked about. That should get you started-good luck!