Are red giants, white dwarfs, and blue giants part of the same cycle?
Yes and no.
The end of a stars life depends on how it began. A blue giant is just a high mass star (10+ solar masses). A low mass star (like our own Sun) will become a planetary nebula and then a white dwarf. A blue giant will become a supernova and then either a black hole or neutron star depending on its initial mass.
When the stars core burns its hydrogen into helium both high mass and low mass stars turn into red giants. But, a high mass stars red giant phases are much larger earning them the name "supergiants."