Stars on the main sequence that are the same size as the Sun begin as yellow stars and turn into red giants as their hydrogen fuel runs out. Other stars shrink or explode, depending on their size.
What happens to a star as it ages is entirely dependent on its size. Stars a quarter of the mass of the Sun do not become red giants; rather, they turn almost immediately into white dwarf stars. Stars half the mass of the Sun become long-lived red dwarf stars. The Sun itself is forecast to turn into a red giant within the next 5.4 billion years, and its radius will be larger than Earth's current orbit. After that, it will begin to fuse helium rather than hydrogen, ultimately forming a planetary nebula and becoming a white dwarf when it has nothing left to fuse. Stars 8 to 12 times the size of the Sun fuse heavier elements, such as carbon and neon, before they become dwarfs.
Stars far larger than the Sun, such as Betelgeuse, do not gradually age. These stars collapse and explode in supernovae, ultimately forming ultra-dense neutron stars or black holes. The supernovae are how elements heavier than iron are naturally formed and scattered.