What is the life stage when a star expands and cools?

During the main sequence of its life cycle, the same stage that our Sun is currently in, a star obtains its energy from nuclear fusion of hydrogen to helium. This nuclear reaction goes on for a very long time, till all the hydrogen in the core of the star is used up. When that happens, the core no longer has a nuclear reaction going on and in absence of the reactions, the core begins to contract. This causes an increase in the core's temperature and a new nuclear fusion reaction starts, in which helium fuses to carbon. This new reaction causes the outer envelope to expand and the star becomes larger in size. This also causes the star to cool down and to appear red. This phase of the star's life cycle, when it has expanded to a very large size and has cooled down, is known as the red giant phase and the star is said to have become a red giant.

The next stage of star's life cycle depends on its mass, with one path leading to white dwarf stage and another leading to neutron star and black hole stages.