First of all you don't really "belong" to a faction, you earn xp for them by completing their challenges, and as your level for each one increases you get more stuff.
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You're mostly correct. You earn reputation for them by equipping their class item- then any time you would normally earn reputation for Vanguard or Crucible (like Bounties, Explore Missions, Crucible games) you [i]instead[/i] earn reputation for the equipped faction.