What is the correct sequence of our Sun's life cycle?
Stars begin their life as a nebula which is a cloud of mainly Hydrogen gas plus some heavier elements.
The nebula collapses under gravity to the point where the centre is hot enough to start Hydrogen fusion. This is a main sequence star. The Sun is currently a main sequence Hydrogen burning star.
When the Hydrogen in the core is used up and the core is mainly Helium, the core collapses. This starts Hydrogen fusion in the layers around the core and the star expands into a red giant. Helium fusion starts in the core when it gets hot enough.
When the Helium is used up the core is mainly Carbon and Oxygen. A star the size of the Sun is too small to start carbon fusion. The core collapses into a white dwarf.
Theoretically after many billions of years, a white dwarf will cool into a black dwarf.