I am puzzled of the “Andromeda–Milky Way collision”, when everything is moving away from the center of the Big Bang?
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The force that drives the expansion of the universe is *very* easily overcome by gravity. The fabric of space is expanding all around us right now, but gravity keeps us together. This is true for scales all the way up to local galaxies, so despite being two million light-years apart, Milky Way and Andromeda are still gravitationally bound to one another, as are all the smaller satellite galaxies in their local group. You don’t see the expansion force take over until you get well beyond that distance.
There is no “center” of the big bang. Everything is moving away from everything else in all directions.
However, the rate to which space stretches is very, very small. Your body isn’t being torn apart by the expansion of space nor are you being thrown off the Earth’s surface.
The other forces acting on you (the electromagnetic forces holding your molecules together, gravity holding you to the Earth) outweigh the effect of the expansion of space by many orders of magnitude. Local forces dominate the situation.
It’s the same on the scale of galaxies. The milky way and andromeda are much farther apart than you and the Earth, which does mean the expansion of all the space between them is a bigger deal. However, they’re also extremely massive. They’re still close enough that gravity “wins” and the galaxies get pulled together. Other, more distant galaxies are more weakly bound, gravitationally, to the milky way and have even more space between them and us. In these situations the expansion of space beats out gravity and those galaxies move away from us.
Things don’t move away from the “center” of the big bang, there’s no center of the big bang. Everything moves away from every other thing. [Balloon analogy](https://www.youtube.com/watch?v=mnENgtCdObo) for further information.
As for how galaxies collide…As you can clearly see, not literally everything moves away from each other. The solar system isn’t expanding nor does the milky way. That’s because of Gravity!
Gravity(and other forces like electromagnetism) gets weaker as the distance increase but the expansion is the opposite, it gets faster the further things are apart. At some point, things are so far apart that the expansion rate won against gravity. On the other hand, if things aren’t too far apart, like between the earth and the sun, or between nearby galaxies, Gravity will be strong enough and the expansion rate will be low enough that it won’t expand or even move toward each other.
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