What do galaxies orbit?
The stars within a galaxy orbit around its center which we now know is a black hole. Most stars exist in pairs with one orbiting the other, our star is an exception although there are some who believe that it does have a red dwarf twin somewhere that we have not discovered. And of course, planets orbit stars.
Most galaxies are moving away from one another and at an increasing velocity. There are exceptions, of course. We know that some time in the future the Milky Way Galaxy and the Andromeda Galaxy will collide and merge into a large new galaxy. This does happen elsewhere in the universe as well.