Let $S$ be a regular surface, and let's consider $\gamma: I \to S$ be a geodesic. Let $ N: S \to S^2 $ be the gauss map. Then $ \beta(s) = N(\alpha(s))$ is a curve $\beta : I \to S^2$ (where $S^2$ denotes the unit sphere). I want to prove that $\beta$ it's also a geodesic, and also I want to know a more general result considering other kind of maps, and not only the gauss map N.
As Ted observes in a comment, if the Gauss map sends geodesics in your surface $S$ to geodesics in the sphere, then geodesics in $S$ are plane curves.
It is a classic result that a (connected!) surface all of whose geodesics are plane curves is in fact (part of) a plane or a sphere.
This last fact is a simple exercise. To start, show that, in general, a geodesic which is a plane curve is a line of curvature. Since in our surface all geodesics are lines of curvature, all of its points are umbilical. Next show that if a surface has all its points umbilical it is a part of a plane or a sphere. These two observations are standard exercises, and should be in pretty much every textbook on the geometry of surfaces.
Any curve $\gamma$ with this property is either a planar line of curvature (such as a profile curve, or meridian, on a surface of revolution) or a generalized helix — a curve on a generalized cylinder that makes a constant angle with the rulings. These curves are characterized by having $\tau/\kappa$ constant ($\tau$ being torsion and $\kappa$ curvature).