The expansion of the universe is happening in all points in our surroundings, all three dimensions at a time. All points in three dimensional space were at the beginning of the universe's one point ( classically, ignoring quantum mechanics) in the beginning of time.
Think of the surface of an expanding balloon, where all points are going away from each other, an ant on the surface would see no specific direction of expansion in its two dimensional universe on the surface. It would be traveling whether it wanted to or not.The expansion on the balloon is not "going" anyplace, for the ant to use it as a transport mechanism. It is more complicated in the four dimensional space time but similar.
There are forces that keep the atoms and molecules and planetary systems from expanding with the space time expansion, but that is another story.