Galaxies collide due to gravatational entanglement, same as more solid objects, as well as simple trajectory bringing them close enough to interact. Andromeda and the Milky Way are likely to collide, or possibly graze close enough for some interaction, sometime in the 2-5 billion year range (opinions vary), so we're going to get a close view of such a thing.
Once two galaxies start to interact, much less outright collide, it does cause stars to shift in their galactic orbits. Perfect spirals become elongated spirals, or the galactic disk warps. A tail of stars begins to form between them as stars, disrupted in their orbits, follow new paths that can lead to stars orbiting a new galaxy, or perhaps thrown off into space all together. This 'tidal tail' is a good clue that a galaxy is currently, or recently has undergone either collision or interaction.
Once the interaction gets closer, the actual collision, it tends to get very bright. The interstellar gas of both galaxies merge with each other, but with differential speeds, which creates eddies and in turn triggers star formation. Actively colliding galaxies are bright with new stars. Interactions with the stars of both galaxies is complex and can definitely throw stars into new orbital trajectories, if not sling-shot right out of the merging galaxies all together.
The form the newly merged galaxies take depends a lot on the sizes of the encountering galaxies and the differential thereof. Dwarf galaxies interacting with large spirals may not do much to the spiral, but two spirals of equivalent size interacting should cause massive disruption of both.
Sometimes a merger is not fully accomplished on the first pass, and a much reduced core of one of the galaxies will emerge from the other side (presumably with the black hole intact), which may continue onwards, or may not escape interaction and circle back for another go some time later. The Omega Centauri cluster, presumed to be in orbit around the Milky Way, just might be such a remnant.
As for the black-holes current models suggest that at least their first contact is likely to cause one of them to recoil as differential rotation is translated into angular velocity, which should cause gravity waves and definitely cause an X-ray flare that should be visible to the right instruments. Mergers of such supermassive black holes are thought to occur, though I don't recall what that would look like to observatories here on Earth.