Is the order of a star's life, red giant, supernova, neutron, and then black hole?
The life cycle of a star is function of its mass, generally given in terms of Solar Mass (the mass of our Sun). The star will continue conversion of hydrogen to helium, that is nuclear fusion, during its normal span of years (which is where our Sun is at the moment). When the hydrogen in the core is finished, the core shrinks; however the outer shell continues expanding due to continued fusion and the star becomes a Red Giant. For stars the size of our Sun, the outer shell is lost and the star becomes a White Dwarf. Larger stars, on the other hand, experiences Supernova explosion, in which extremely large amount of energy is released in very short span, lighting up that section of galaxy for few days. For stars up to 3 solar masses, the supernova will leave behind a Neutron star, while more massive stars will convert into a space of infinite gravity, known as the Black Hole.
Thus, the life cycle of a star is a function of its mass. In summary,
For our Sun: Star -> red giant -> white dwarf
For 1-3 solar mass: star -> red giant -> supernova -> neutron star
For > 3 solar mass: star -> red giant -> supernova -> black hole
Hope this helps.