Do stars of a galaxy change their positions relatively to each other?
Stars do in fact move relative to one another within galaxies of all types. The orbital period of stars in a typical spiral galaxy (at around the same distance as the Sun is from the center of the Milky Way) is on the order of hundreds of millions of years. For the Sun it's something like 230 million years (source). A particularly old person (~100 years old), will have been around long enough to see something like one-tenthousandth of 1% of the orbit of stars in this region of the galaxy! Most of these stars simply do not move much relative to us in that amount of time.

The exception to this view of the motions of the stars is the Sagittarius A* (which is not a star at all, but is in fact a supermassive black hole at the center of the Milky Way). The period of stars traveling around it are on the order of 10's of years.



Also, to give you a rough idea of how galaxies look dynamically, here's a link to a particularly nice galaxy simulator implemented in the browser (created by Adrian Price-Whelan).