Will there ALWAYS be a potential drop (negative potential) whenever you go across a resistor?
Yes, because if there wasn't, you wouldn't call it a resistor.
With no drop and no change at all, the component would simply be called a conductor.
The whole point in resistors is to control current by limiting and reducing it in fitting amounts. It does this by reducing the voltage, which is the driving force that causes current.
I know that the direction of current is from positive to negative, and that the direction of voltage is simply the opposite of current (from negative to positive).
It doesn't really make much sense to talk about "direction of voltage". Voltage is not a flow like current. Two points can have different potentials, and the voltage is just found by subtracting one from the other. Voltage is just difference in potential.
A difference is always end-situation minus start-situation. Naturally, since current runs from high to low potential, the end-situation is at a lower value of potential than the start-situation, and so the difference - the voltage - will be negative.
Across other components, like e.g. a battery or other voltage source, the end-situation (the positive terminal where current leaves from) will be at a higher potential than the start-situation (the negative terminal where current arrives at). The voltage would here be positive. But the point is not these signs - the only important thing is what the signs are compared between components (battery voltage will be opposite in sign to resistors, always).
You can call this a "direction" if you want (though it hurts my ears), just know what voltage actually means. Nothing flows.
My question is - whenever you have a resistor, will there ALWAYS be a potential drop? Or, will your sign and direction of voltage across a resistor depend on if you are going from positive to negative (negative potential -- potential drop), or negative to positive (positive potential)?
As mentioned, there will always be a drop, yes. Simply because voltage is the difference in potential energy, and energy is being lost as heat (on purpose) in resistors, there will always be a lower potential where the current exits the resistor than where they entered. Regardless of current direction, energy will be lost in a resistor.
Were the current reversed, there would still be a loss, and the potential would still be lowest at the end where current exists the resistor.