Is it true that galaxies are moving away from us and into the vast, empty space of the outer universe beyond the Big Bang?
Yes, sort of true but your question is a little wrong.
All of the galaxies and the stars in them have been expanding away from each other since the Big Bang almost 13.7 billion years ago. But they are not moving away ... into the vast, empty space as you imply - rather the galaxies and space/time itself are actually expanding, but not into "anything".
One analogy is like raisins in a bread or pudding that is being baked - the raisins (or galaxies) and the bread (space-time itself) are expanding together and therefore the galaxies (raisins) are moving away from each other, but not "into" empty space. The analogy is only rough, as you could argue that the bread is expanding into the air in the kitchen - but this is not the case with the universe.