I agree with Tom and Ryan that it is worthwhile to learn proofs of the classification of surfaces. I think that the result get a bit of a bad rap since the "standard" combinatorial proof that everyone used to learn (which appears in Seifert-Threlfell's book and Massey's book) is complicated and unenlightening. However, there are now a number of nicer proofs available. Here are a few of my favorites.
1) If you like Morse theory, there is a nice proof in Hirsch's book on differential topology.
2) There is a slick combinatorial proof in Armstrong's book "Basic Topology". I believe that this is the source for the proof mentioned above by Tom Church that Benson Farb likes to give.
3) In Fomenko-Matveev's book "Algorithmic and Computer Methods for Three-Manifolds", there is a nice proof using handle decompositions.
4) There is finally John Conway's "ZIP proof", which was written up by Francis and Weeks in their paper "Conway's ZIP Proof".
All of these proofs assume that the surface has been equipped with either a triangulation (for numbers 2-4) or a smooth structure (for 1). For nice approaches to this, see the answers to my question here. However, when you are first approaching these types of results, I would recommend just assuming that the surfaces can be triangulated or smoothed.
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