In the vast, vast majority of cases, not much happens. Galaxies are a whole lot of empty space.
If the Sun was a soccer ball in a field in California, the nearest star of similar size, Alpha Centauri, would be another soccer ball all the way across the continent, in Greenland. This is a scaled view, where everything is proportional. Think of all that empty space in between.
So the galaxies would merge, or keep swirling around each other, but for the most part stars and their systems would just continue as they were. It is extremely unlikely that any given star would collide. That's true for most of the volume of the galaxies.
In the center, however, things are a little different. Most galaxies already have very massive black holes at the center. These may collide and merge, or orbit each other a very long time, or continue separately.
The formation of new black holes as a result of galactic collisions must be a very rare event. Even then, the entire galaxy would not "become a black hole"; rather, a small fraction of it would be added to an existing black hole or, in very rare cases, produce a new black hole.
For most observers in such galaxies, there would not be dramatic changes, even over very long time intervals. This is because, again, galaxies are mostly empty space, and because the stars you see with the naked eye are very close to you, within a few dozen to a few hundred light years, which is a small portion of the whole galaxy.
EDIT: As shown in the comments, it's more accurate to think of Alpha Centauri as not a single soccer ball, but instead a larger soccer ball and a volleyball separated by a few hundred meters - it's a double star. Then Proxima Centauri would be a pingpong ball hovering 100 km away from Alpha.