Heroes see quests based on threat, and if it matches their "desires" they'll decide "I want to do that!" If no such quest is around, they'll do some relatively default action, such as training, researching a spell, hunting down monsters, etc - so now let's say they see a Quest they want to do. They see its "Danger Values" (and this is the public number, anything you've done to increase the danger is a nasty surprise) when they get there the complexity of the quest gives them X number of dangers + any dangers you've put there. So they may have "getting lost" "battle with monster" etc. This pretty much all goes on behind the scenes though yo ucan see the list of Dangers they faced - Heroes in battle can get wounds, decreasing their hitpoints. Hit points ONLY COME BACK by resting for multiple turns in cities or villages, so hp loss at least can slow them down, and if they lose enough they can get a permanent (relatively) injury. If a hero feels like the Quest is too hard, but still wants to do it - they will try to either "Strengthen themselves" or "Find an Adventuring Group" to help - if neither of those will do they'll just wander off disconsolate with a -1 to their spirit. Hero personalities play heavily into how an Adventurer pursues quests.
Campaigns are multi-stage quests that have specific requirements, but they also have an added component of "level of victory" - if the heroes beat all 5 quests in a large campaign they get the best ending. They fail all 5 they get the worst ending (yay for you).
When you do something to destablize it adds as Condition with a type, types stack and increase one another - generally they have a timer where they will go down, but others have an "until X occurs" condition for them to end. Corruption of a leader or to gain influence depends on the type - some of them are "Expended", some of them are permanent unless heroes undo it, some decrease over time. The easier it is to get the influence generally the faster it will go away, whereas very complex actions will result in permanent effects.