To Kevin's excellent list I would add Guillemin and Pollack's very readable, very friendly introduction that still gets to the essential matters. Read "Malcolm's" review of it in Amazon, I agree with it completely.
Milnor's "Topology from the Differentiable Viewpoint" takes off in a slightly different direction BUT it's short, it's fantastic and it's Milnor (it was also the first book I ever purchased on Amazon!)
I wouldn't describe it as an "area" of math, exactly, but there are certainly mathematicians who study embeddings of graphs into surfaces and related objects. Let me recommend two sources to learn more about what people are doing. I am not familiar with the book you listed, so I don't know if it sufficient preparation for reading them; however, I am a big fan of jumping in to an unknown field and then learning background information "as you go".
The first is the long book
MR2036721 (2005b:14068)
Lando, Sergei K.(RS-IUM-M); Zvonkin, Alexander K.(F-BORD-LB)
Graphs on surfaces and their applications.
With an appendix by Don B. Zagier. Encyclopaedia of Mathematical Sciences, 141. Low-Dimensional Topology, II. Springer-Verlag, Berlin, 2004. xvi+455 pp. ISBN: 3-540-00203-0
This might be the closest to what you are looking for in that it focuses on connections with algebraic geometry and number theory, especially Grothendieck's theory of dessins d'enfants and questions related to the moduli space of curves.
The second is a very different direction, namely the study of "spatial graphs"; ie embeddings of graphs into 3-space and other 3-manifolds. This is basically a generalization of knot theory. One possible way of getting into this subject is to start with the survey
MR2179645 (2006e:57009)
Ramírez Alfonsín, J. L.(F-PARIS6-CM)
Knots and links in spatial graphs: a survey. (English summary)
Discrete Math. 302 (2005), no. 1-3, 225--242.
Best Answer
If you just want to get a feeling for invariant theory, here are some books that aren't necessarily comprehensive but nevertheless are enlightening at a more leisurely pace as compared to GIT, which would be useful for someone who isn't as familiar with algebraic groups and algebraic geometry:
Santos and Rittatore - Actions and Invariants of Algebraic Groups: Minimal prerequisites. A very gentle introduction to some aspects of invariant theory, including some motivation via Hilbert's 14th problem. This book also contains most of the required theory of linear algebraic groups.
Dolgachev - Lectures on Invariant Theory: This takes a more geometric viewpoint and might be something you are interested in. This only requires some basic knowledge of algebraic geometry.
Schmitt - Geometric Invariant Theory and Decorated Principal Bundles: this might also be interesting if you are interested in the geometric applications and the related geometry, though I haven't looked into this book very much, but Part 1 does contain a fairly leisurely-looking introduction to GIT
There is also Popov's and Vinberg's treatise "Invariant Theory" in the Ecyclopedia of Mathematical Sciences Volume 55 (Springer) which contains a good summary of the classical results in characteristic zero.