Marco, do you mean the space of (constant-length) parametrized geodesics or the space of geodesics. In the first case you would get the tangent bundle, as Ryan said. The second case is in general more complicated, and I think you won't get a nice manifold structure on that space in general. Nice would mean for example that the projection from the tangent bundle (space of constant lenght parametrized geodesics) to the space of geodesics is a submersion. This is not the case for a flat torus. More generally, this is not the case when there exist a geodesic $\gamma$ such that the closure of the image of its tangent curve $\gamma'$ is not the image itself. I am not sure about this, but I guess that this will always be the case on compact manifolds such that not all geodesics are closed.
But if you assume for example that you have negative curvature on a simply connected manifold, you get a nice manifold structure, as you may see from the theory of Jacobi fields for geodesics.