Everything in the universe appears to be moving away from everything else. Think about it this way. Take a balloon before you blow it up and stick a few stickers on it. Then blow up the balloon. All the stickers look like they're moving away from each other. That's what the universe is like - everything is moving away from everything else in the expansion. But you're not expanding, and the galaxy doesn't seem to be, because local gravitational forces are stronger than the expansion forces.