The factions operate as a timer for the game and as a source of influence points. Whenever one of the quests has the symbol and a "+" sign, that symbol moves the number of plus-signs down the track. If it reaches the final slot, the game is over... if any player has enough influence to declare victory they win, otherwise the faction that got to the end wins and all the players lose.
The agenda cards you draw are random, so I think about it more as discovering who my character is vs. deciding who my character is... you woke up from <x> and are trying to survive, finding more and more about your beliefs (what "drives you"... a.k.a. the agenda cards) as you proceed. As you find out more and more about yourself, that will guide the decisions you make.
Discovering the Vault tiles does not trigger anything until the result of a quest or encounter card adds a card to the encounter deck that has that vault number... that's when you can start interacting with it (by performing an Encounter action on the Vault space in order to draw from the corresponding Vault encounter deck). Until then, you haven't discovered it.
EDIT: Didn't like my wording