Guild Wars 2

When do I start getting them for the main story, and not just my personal one?

The first story mode dungeon is ascalonian catacombs at lvl 30, the dungeon is in plains of ashford (the first charr map).
The story mode dungeons are not for the main story, they are just a side story about the Destiny’s edge “reunion”, when you meet then at the first time in Lions Arch you learn that they are angry at each other, and during the story mode dungeons you help then be friends again, until the final mission which is Arah story mode (that can be done solo now) where you fight Zaithan with their help.