I am not sure what class of surfaces or topological spaces this is a theorem for but it should at least include the plane, circle, sphere and torus (hopefully also the Klein bottle) - so part of the question is what class of surfaces to consider:
Given a fixed base point consider every path in some homotopy class. How can one prove that a shortest path exists and that it is a geodesic?
On a complete Riemannian manifold, there is always a shortest geodesic in a given homotopy class of loops based at a point. See e.g. here which treats the (more general) case of Finsler manifolds. (I found this particular reference just by googling, and you should be able to find many other references in the same way.)