This seems unintuitive, but the Universe doesn't appear to have a center for galaxies to move away from. This is because the Big Bang seems not to be a single event at a particular place, but instead an event at every place. In other words, the Big Bang seems to have happened at every point in the Universe, possibly infinitely in all directions (note that "direction" didn't actually mean much back then). Because of this, there is no "center" for the Universe to spread from.
Instead, when it is said that the Universe is expanding, it is meant that the fabric of space-time itself is literally getting bigger. This is thought to be fueled by dark energy, but the mechanism is still unknown. In other words, we don't know why it happens.
This gives us the odd situation that two galaxies that are stationary relative to one another still end up "moving" farther away because the fabric of space in between them is getting bigger. Even two galaxies that are moving towards each other relative to some other object still end up receding from one another if they are far enough away, because the space in between them is getting bigger faster than they can traverse it.
Even light, if its source is far enough away from the observer, will fail to traverse a sufficient distance, as the space between the light's source and the observer will literally get bigger as time goes on. We already have one limit on how much of the Universe is observable, that being the distance beyond which light has had time to travel since the Universe began. The expansion of the Universe adds a second limit, which is the distance beyond which light will never reach us due to the space in between us and its source expanding faster than light could ever traverse it.
I am not particularly knowledgeable on this subject and may have made mistakes. Anyone who is more knowledgeable should feel free to offer corrections to this answer.