I recently made a level 8 Storm Herald Barbarian (from XgtE) with the Desert aura. The level 6 feature allows me to set flammable objects that are not being carried or worn by anyone else on fire. So my question is this. If I set a quarterstaff completely on fire, how much extra fire damage would it deal to my opponent, how much to me, and how long would it last before burning?
It's up to your DM
Unless the ability that sets the items on fire says how it changes the damage of effected weapons, I'd expect the baseline might be for it not to change anything at all. Anything more is up to your GM to homebrew for you.
According to the Torch entry in the Player's Handbook, hitting somebody with a lit torch does one point of fire damage. A burning quarterstaff is unlikely to be much more effective at delivering fire damage, especially if the flames are effecting the wielder as much or more than they are the target of the attack. Maybe you'd still do normal quarterstaff damage in addition to a little fire damage, but it could be easy to imagine the flames effecting your ability to swing it effectively, so you'd do less damage overall.
As for how much damage you'd take yourself, that's also up to the DM to decide. I'd guess it would not be any worse than the 1d8 damage dealt by the Create Bonfire cantrip, but your DM might think it's more dangerous to be holding the burning object than just to be standing among some flames. You will have resistance to the fire damage, from the same class feature that lets you start the fire. But don't be shocked if you end up taking more damage than your enemy anyway! Fire is not an easy or safe weapon to wield without a lot of magical control. And "control" is not really something Barbarians are known for.