The Earth, while spinning (rotating) is traveling “sideways” as it revolves (orbits) around the Sun. Our axis is slightly tilted in relation to the plane of our orbit (hence, the seasons). Our axis always points, just about, to the North Star, Polaris. Of course there are many bodies in our solar system each with its own rather fixed path. But our solar system, with all its moving parts, is headed toward where Leo currently is (because our solar system supposedly is circling the Milky Way which itself is moving towards where Leo currently is). We can say our solar system is “circling” the Milky Way Galaxy because they tell us that we are on an outer edge of the Galaxy and that the Galaxy is spinning. But, the Andromeda Galaxy is in the path of the direction we are moving, so we will collide with it before we reach whatever will be in the direction of where Leo currently is.
But there is so much space between stars that spiral galaxies colliding have different planar momentums and become giant eliptical galaxies. These have no spiral shapes, but become football shaped with no regular orbital pattern. One would expect head-on collisions of stars with the random orbits which could be in opposite directions resulting in tremendous collisions.