------- "What are the most important stages of moral development?"
I am considered to be an expert in Lawrence Kohlberg theory of moral development and stages. As far as I know, the most important stages of moral development were conceptualized by the developmental psychologist mentioned above, Lawrence Kohlberg.
His theory elaborated at length on six moral stages: Stages 1, 2, 3, 4, 5. and 6.
It is beyond the scope of this post to elaborate on these stages. Even so, it should be said that these stages are hierarchical and integrative. I would suggest you to read Kohlberg's book titled The Psychology of Moral Development (1984).
He fist conceptualized six "hard" structural stages in the book mentioned above, which has 729 pages. Because of the rarity of Stage 6 as a structural stage, in 1987 Kohlberg's Stage 6 was subsumed under his Stage 5. However, in 1990, he co-authored a book's chapter titled " The return of Stage 6: Its principle and moral point of view. In T. Wren (Ed.), The moral domain (pp. 151-181). Cambridge: The MIT Press. More to the point, Kohlberg still elaborated on a non-structural stage, Stage 7.