If all distant galaxies are expanding away from us, does this mean we're at the center of the universe?

No

Explanation:

Is a thing of relativity and perspective, let me put it this way: if you have lived all of your life in an island and you only see water around you, you will feel that you are in the middle of the sea, the farthest you can see is always the same distance and if you look at the same distance in all directions you will be at the center in the end.

Not all galaxies are expanding away, andromeda is in course to collide with the milky way. We are going to be in the same space that Andromeda some day! but no worries, it's really improbable that even a star from Andromeda crash with a star of the milky way, the distance among those stars is huge!

We are not in the center of the universe, we look in all directions of the universe and how we can see the same distance it might feel like we are in the center, but it's not necessary true.