What happens first when a star begins to run out of fuel? The star's core shrinks. The star becomes a black hole. The star becomes a white dwarf. The star's outer layer drifts away.

When a star begins to run out of fuel the first thing that happens is that the star's core shrinks.

Explanation:

Any star only has a finite amount of fuel that it burns throughout its life and when the fuel starts running it inside the core of the star its core begins to shrink as there is not enough material to begin a reaction from in the star.

This is generally what is followed then by the star becoming a white dwarf as the outer layer gets stripped away from the weaker core which keeps on shrinking and then the star eventually bursts to become a supernova or it just becomes a black hole.