For example, I am vulnerable to fire and have an item that gives resistance to fire. Do the two cancel each other out, or does one supercede the other?
No they don't, but you do get a 1hp bonus if the incoming damage is an odd amount.
"Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater. This is the case in every circumstance unless a more specific rule (see below) specifies that you should round up instead. If any doubt emerges, round down!"
"Resistance and then vulnerability are applied after all other modifiers to damage."
So you have a tiefling with fire resistance, but a curse or effect that gives fire vulnerability. Now you take 25 incoming fire damage. This is first halved from resistance 25/2=12.5 rounded down to 12 and doubled back to 24 for the vulnerability. If the incoming damage was an even amount say 24, it would be halved to 12 then doubled back to 24.