If we look at Newtonian physics, and how galaxies will interact, a central black hole should just be considered as a massive, dense object.
The Milky Way does not fall into its own central black hole, it orbits it - as physics tells us it should.
When we get closer to Andromeda, the gravitational influence of Andromeda will act more strongly on us, and when we get really close, individual masses within each galaxy will have dramatic effects on each other, but as with any such system, the two black holes will not suck everything in.
If the two galaxies end up coalescing - which is not a given - the orbits of the stars and the black holes will be very complex. For billions of years the 2 black holes will orbit each other, getting closer as they shed energy, but during that time the stars around them will suffer many effects, including:
- some will be expelled
- some will hit the black holes
- some will go nova
- and so on
The best way to understand what will happen is to avoid thinking too deeply about black holes being weird, and treat them as dense masses. For most purposes this will help you model galactic collisions.
Have a look at this NASA simulation of the collision between the Milky Way and Andromeda: