Actually most galaxies are holding relative position but the space in between the galaxies is expanding. Just think of it as the universe being a huge body hosting the galaxies and the space in between is growing with the host (and spontaneously creating dark matter).
Most of the other galaxies are accelerating away from each other..Except for the Andromeda Galaxy,which is on a direct collision course with earth.3 billion years from now our galaxy goes poof.
It is a little more complicated than that. Most galaxies are falling into their local groups which is actual movement through space. Andromeda and the Milky-way are falling toward each other, as well as a few other small satellite galaxies around us. This bunch in turn falls toward our local group. This is the scaling of structure in the universe. Close galaxies eventually merge, not destroyed. This is a fairly "violent" event, but the stars do not collide, or destroyed, the environment will endanger life though. Actually, mergers usually lead to a renewed episode of rapid star formation.
The Hubble expansion is space being stretched that you described. Take a piece of stretching material and draw a few points on it - then stretch it. The distance between the points increase even though the ink spots cannot move on the sheet. If you put ants instead of ink, the ants will walk about while you stretch and this is the superposition of the two "speeds".
The nature of a stretch is that the farther a way a point is from you, the faster it will move away from you - the distance it gains is the sums of all the stretches between you and all the other points along the way. This means that the cosmological expansion dominates over large distances and essentially all distant galaxies move away from us. For nearby galaxies, the "falling" speed will dominate over the stretch and we may see them even coming toward us.
What is created in the added space is dark "energy", not dark matter. While dark energy tries to make the universe "shrink" (attracts by gravity just like normal matter), dark energy is the equivalent of "pressure" that makes the universe expand. The nature is this energy is completely unknown. The space density of the energy (the pressure) is taken to be constant for lack of better information - this is referred to as "Einstein's cosmological constant" if you heard the term.
