I'd say that her main story arcs are two: her Kalos and Galar arcs. She has some other arcs but those are kind of a way to set a basis for the two main arcs.

Kalos arc: Briar meets Lysandre, joins Team Flare and becomes the Commander. After a series of unfortunate events, Lysandre dies in Geosenge Town after the XY Protag stops his plans, and Briar ends up becoming the Boss of Team Flare briefly before disbanding the whole team. After this, Briar moves back to Unova to escape the Interpol agents who were still searching for Lysandre's allies.

Galar arc: Rose travels to Unova to search for a genius scientist he heard about in Galar that could help him with an important project of his. Although Briar had declared she was done with researches, his project interests her enough to accept working as a scientist again. This project would be known as Eternatus some years later. Her main arc.