What the achievement asks
Reach Revered with six reputations. Ten points, account-wide, both factions.
The six are not the same on both sides. Four of them are your faction's own, and two are shared.
The three zone reputations come from the zone you quest through, the fourth from the war campaign, and the last two from work spread across both continents.
Questing gets you most of the way
Finishing a zone's story carries its reputation a long way towards Revered on its own, so a character that has done the three zone stories is usually close on four of the six.
What is left over is the two shared reputations, which have no zone of their own and therefore no questline to carry them.
Island expeditions are the shortcut nobody mentions
Each reputation has a turn-in item that expeditions can drop, and they are worth a large chunk each. Tortollan Seekers in particular is the reputation people stall on, and its soggy pages come out of islands rather than out of Kul Tiras or Zandalar at all.
That makes this achievement overlap neatly with Expert Expeditioner, which wants fifty Mythic island wins. Anyone running those is farming reputation whether they mean to or not, and the turn-ins are the fastest route to Revered on the two that questing does not touch.
Emissaries are the rest of it
Beyond that it is world quests and the emissary that rotates through them. Both continents count for both factions, so take whatever emissary is up rather than restricting yourself to your own side's zones.