Anne gets to come of age in somewhat quieter fashion during season three, since she's not the primary heir. She has a fling or two, bonds with her grandmother Princess Alice, and tries to support her brother Charles where she can. It's Anne's revelation of the whole mess between Charles, Andrew, and Camilla, however, that pushes the Queen over to the side of "break them up."
Notably, season three skips over some major events in the real Anne's life that took place during the corresponding years. She married fellow equestrian Mark Phillips in 1973, was the victim of a kidnapping attempt in 1974, and competed in the Olympics in 1976; none of those events have were shown during season three, which nevertheless ran up to 1977.