If this has been answered already, please let me know and I'll delete the question.
ADDED: I'd prefer to assume a smooth structure, rather than a triangulation.
If this has been answered already, please let me know and I'll delete the question. ADDED: I'd prefer to assume a smooth structure, rather than a triangulation. 


A very simple and low tech solution to this problem is the very first proof, given by Möbius in 1863. He assumes that the surface is smoothly embedded in $\mathbb{R}^3$, and slices it by a family of parallel planes. Assuming that the orientation of the planes is general and that they are sufficiently close together, this cuts the surface into simple pieces  either disks, annuli, or pairs of pants. It is then quite easy to show that the result of assembling such pieces is always a sphere with handles. 


Count the number of disjoint nonseparating embedded circles. 


Consider harmonic functions $f$ with exactly 2 logsingularities of weight $\pm 1.$ (locally $f(z)=a log\parallel z\parallel+g,$ $g$ being smooth at $z=0$, $a$ being the weight) on your compact surface equipped with a Riemann metric. They exists by standard elliptic theory ( the two weights $a_1$ and $a_2$ have two add to zero). Consider $\partial f,$ the complex linear part of the differential. This is a meromorphic section of your canonical bundle. Then $deg\partial f=\frac{1}{2\pi i}\int KdA,$ as the LeviCivita connection defines a complex linear connection on the canonical bundle. This shows that if the total curvature is large enough $\geq 4\pi,$ $f$ will not have critical points (only two singularities). Moreover $e^f$ is the real part of a holomorphic bijection onto $CP^1.$ If $f$ has a critical point, then you can easily construct a nonsepreating loop, as in Morse theoretic proofs. You cut your surface, and add two disc (with the right orientation). One, can easily see, that this must increase the total curvature by $4\pi,$ and you end up with the twosphere after a finite number of steps. 


Check out this nice paper by Thomassen for a short selfcontained proof. 

