Moise, "Geometric topology in dimensions $2$ and $3$" is perhaps what is closest to what you are looking for.
Let me mention three other references.
P. Buser, "Geometry and spectra of compact riemann surfaces" proves the following strengthening of the triangulation result:
Any compact Riemann suface of genus greater than two admits a trianugulation such that all trigons have sides of length less than log(4) and area between 0.19 and 1.36.
There is a book by Munkres, "elementary differentiable topology", that proves the more general fact that any smooth manifold is triangulable. It's long (it takes more or less the whole book) but it is really interesting.
Also the proof of the classification of smooth compact surfaces using Morse theory, in the book of Hirsch, "differential topology", is nice (but here again, it does more than just showing that compact surfaces are triangulable).