Fate seems to rely on people putting forward ideas and seeing if the GM rolls with it. Everyone tells the story, really, not just the honcho with the hat.
As for combat- There's four main actions in the whole game.
(This is just a quick catch-up for those who haven't got the time for the CRB)
Overcome an obstacle (Using your skill to solve a miscellaneous problem. Basically using burglary to open a window, or empathy to calm a crowd)
Create an advantage (Using your skill to either create an aspect with a free invoke, or to put a free invoke an an aspect which already exists)
Attack (Hit and deal damage to the enemy based on the amount of positive shifts. This damage is then either transferred into stress or consequences- or they're out of the fight.)
Defend (Roll to stop bad things happening to you. This being anything from attackers, mind control, or getting burnt by a fire)
So the bottom three are typically the only ones used in combat.
Current combat, if you wanted to play with fire, you could say
"A university must have a bunch of candles and lanterns around, I'd like to use my shoot skill to create an advantage... Say, the 'Burning Oil' aspect."
Or you could attack it straight, and if it takes enough damage it may have to buy off its damage by developing 'frayed', 'tattered', or 'burning' aspects.
So whatshisscholar I assume has just run off to get water so that he can put the 'sodden' aspect on the paper demon. Thus something something combat.
Unfortunately this means if someone wants to invoke an aspect which someone else has created, then the GM might need to post many times a turn.
But Cinder currently has a pretty good handle on it.